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 |
|---|---|---|---|---|
47,143 | <p>I want to create a table of replacement rules. </p>
<pre><code>g[a_, b_] := a -> b
t1 = Table[10 i + j, {i, 5}, {j, 3}]
t2 = Table[ i + j, {i, 5}, {j, 3}]
g[ # & @@@ t1, # & @@@ t2 ]
</code></pre>
<p>The correct output is below:</p>
<pre><code>{{11 -> 2, 12 -> 3, 13 -> 4},
{21 -> 3, 2... | gwr | 764 | <h2>Excursion: Using Modelica within WL</h2>
<p>I would like to mention this possibility here as importing and running <a href="https://specification.modelica.org/master/" rel="nofollow noreferrer">Modelica</a> has since been implemented within the <a href="https://reference.wolfram.com/language/guide/SystemModelingOve... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| amWhy | 9,003 | <p>The probability is, in fact, $\large\frac14$.</p>
<p>Wherever the first point is chosen, the diameter on which it lies (the diameter being determined by the circle center and the first chosen point) divides the circle into two symmetric semi-circles, so the second and third points (assuming they are distinct) must ... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Linos | 553,526 | <p>The triangle must be acute if to contain the center.
Then, if say X is the biggest angle of the triangle, we need to calculate
$P(X<90|X>60)=
P(60 <X <90)÷P(X>60)=
30/180÷120/180=
\frac{1}{4}$.</p>
|
250,484 | <p>For a fixed set $X$ and a finite collection $E_1,E_2,\ldots,E_k\subseteq X$, define the binary relation <em>adjacency</em> as follows: $E_i,E_j$ are adjacent
if their intersection is nonempty.
We term the transitive closure of this relation by
<em>transitive adjacency</em>
and define the <em>adjacent union</em> by
$... | Vinicius dos Santos | 8,193 | <p>You can think in terms of <a href="https://en.wikipedia.org/wiki/Intersection_graph" rel="nofollow noreferrer">intersection graphs</a>. The transitive adjacency tells you when two vertices are adjacent. The adjacency union is then empty if and only if two of $E_1,\ldots,E_k$ are in distinct connected components.</p>... |
1,906,332 | <p>I know every complex differentiable function is continuous. I would like to know if the converse is true. If not, could someone give me some counterexamples?</p>
<p><strong>Remark:</strong> I know this is not true for the real functions (e.g. $f(x)=|x|$ is a continuous function in $\mathbb R$ which is not different... | Thomas | 128,832 | <p>No, of course not. Complex differentiability is a very stringent property.</p>
<p>For examples, just look at any pair of continuous but not (real) analytic functions $a,b:\mathbb{C}\rightarrow \mathbb{R}$ and let $F= a+ib$.</p>
|
2,472,313 | <blockquote>
<p>Suppose $x$ and $y$ are real numbers and $x^2+9y^2-4x+6y+4=0$. Then find the maximum value of $\left(\frac{4x-9y}{2}\right)$</p>
</blockquote>
| Raffaele | 83,382 | <p>Lagrange multipliers method</p>
<p>Maximize
$$k \left(x^2-4 x+9 y^2+6 y+4\right)+\frac{1}{2} (4 x-9 y)$$</p>
<p>$
\left\{
\begin{array}{l}
k (2 x-4)+2=0 \\
k (18 y+6)-\frac{9}{2}=0 \\
x^2-4 x+9 y^2+6 y+4=0 \\
\end{array}
\right.
$</p>
<p>$$\left(x=\frac{6}{5},\;y=-\frac{2}{15}\right);\;\left(x=\frac{14}{5},\;... |
1,972,002 | <p>Show an open set $A \subseteq \mathbb{R}$ contains no isolated points.</p>
<p>My attempt:
well if we can show an arbitrary element $a \in A$ is a limit point then we will be done. So since $A$ is open, there is an $\varepsilon > 0$ such that $B_\varepsilon(a) = (a - \varepsilon, a + \varepsilon) \subseteq A$. To... | RJM | 376,273 | <p>Hint: About every point must exist a neighborhood of radius greater than 0 that is a subset of A. Take an arbitrary point ,x, and neighborhood, B, of radius h, such that B $\subset$ A. Let N$\epsilon \gt$ h, where N $\in \mathbb{N}$ and $\epsilon \gt 0$. You can find a point of B for n $\ge$ N in each smaller neigh... |
3,956,392 | <p>So the question is as follows:</p>
<blockquote>
<p>An urn contains m red balls and n blue balls. Two balls are drawn uniformly at random
from the urn, without replacement.</p>
</blockquote>
<blockquote>
<p>(a) What is the probability that the first ball drawn is red?</p>
</blockquote>
<blockquote>
<p>(b) What is the... | Allawonder | 145,126 | <p>The second probability is</p>
<p><span class="math-container">$$\frac{m-1}{m+n-1},$$</span></p>
<p>which is not the same as the first, so there's nothing to be worried about.</p>
|
23,937 | <p>I am trying to use the <code>Animate</code> command to vary a parameter of the Lorenz Equations in 3-D phase space and I'm not having much luck.</p>
<p>The equations are:</p>
<blockquote>
<p>$\begin{align*}
\dot{x} &= \sigma(y-x)\\
\dot{y} &= rx-y-xz\\
\dot{z} &= xy-bz
\end{align*}$</p>
</blockquote>... | halirutan | 187 | <p>You can always define the recursive function yourself and use <em>memoizing</em> to speed up computation:</p>
<pre><code>g[n_, 0] := g[n, 0] = n;
g[n_, 1] := g[n, 1] = n^2;
g[n_, k_] := g[n, k] = g[n + 1, k - 1] + g[n + 2, k - 2];
Table[g[n, k], {k, 0, 10}, {n, 0, 10}] // TableForm
</code></pre>
<p><img src="http... |
23,937 | <p>I am trying to use the <code>Animate</code> command to vary a parameter of the Lorenz Equations in 3-D phase space and I'm not having much luck.</p>
<p>The equations are:</p>
<blockquote>
<p>$\begin{align*}
\dot{x} &= \sigma(y-x)\\
\dot{y} &= rx-y-xz\\
\dot{z} &= xy-bz
\end{align*}$</p>
</blockquote>... | Vitaliy Kaurov | 13 | <p>Note, this can be solved in general form. Start as</p>
<pre><code>RSolve[{G[n, k] == G[n + 1, k - 1] + G[n + 2, k - 2]}, G[n, k], {n, k}]
</code></pre>
<p><img src="https://i.stack.imgur.com/q71oO.png" alt="enter image description here"></p>
<p>You have two unknown functions C(1)[x] and C(2)[x] that you can find ... |
2,372 | <p>How can I find the number of <span class="math-container">$k$</span>-permutations of <span class="math-container">$n$</span> objects, where there are <span class="math-container">$x$</span> types of objects, and <span class="math-container">$r_1, r_2, r_3, \cdots , r_x$</span> give the number of each type of object?... | Community | -1 | <p>There are 20!/4!5!8!3! arrangements of the 20 alphabets. Consider the action of $S_5$ on the arrangements by permuting the last 5 digits. You want to compute the number of orbits. You can then try to do it by <a href="http://en.wikipedia.org/wiki/Burnside%27s_lemma" rel="nofollow">Burnside's formula</a>. It is not d... |
2,015,809 | <blockquote>
<p>If $$f(x)=3 f(1-x)+1$$
for all $x$, what is the value of $f(2016)$?</p>
</blockquote>
<p>I am not sure how to do this, because I see two "$f$"s.</p>
<p>All I could try is substituting,</p>
<p>$$f(x)=3(1-2016)+1\\
=-6044$$</p>
<p>Which I am pretty sure wrong.</p>
<p>How do I deal with this quest... | Hermès | 127,149 | <p><strong>Hint:</strong> $f(2016) = 3f(-2015) + 1$. But $f(-2015)= 3f(2016)+1$. Conclude</p>
|
2,015,809 | <blockquote>
<p>If $$f(x)=3 f(1-x)+1$$
for all $x$, what is the value of $f(2016)$?</p>
</blockquote>
<p>I am not sure how to do this, because I see two "$f$"s.</p>
<p>All I could try is substituting,</p>
<p>$$f(x)=3(1-2016)+1\\
=-6044$$</p>
<p>Which I am pretty sure wrong.</p>
<p>How do I deal with this quest... | fleablood | 280,126 | <p>f (2016)= 3f (-2015)+1</p>
<p>f (-2015)=3f (1-(-2015))+1=3f (2016)+1.</p>
<p>So f (2016)=3 (3f (2016)+1)+1=9f (2016)+4</p>
<p>So -8f (2016)=4</p>
<p>So f (2016)=-1/2.</p>
<p>Now the real question, is how to solve for <strong>all</strong> x.</p>
|
1,519,251 | <p>This question was in school maths challenge.
I dont know how to approach this one.. any help would be appreciated.</p>
| fleablood | 280,126 | <p>Ends in 100 zeros means that $5^{100}$ is a divisor. If $n \ge 5^im$ then $n > 2^im$ so if $5^{100}$ is a divisor so is $2^{100}$ so is $10^{100}$. so $5^{100}$ divisor is <em>sufficient</em> to end in 100 zeros. </p>
<p>To end in <em>exactly</em> 100 zeros means that in {1,2, ...., n} there are precisely 100 o... |
275,527 | <pre><code>Sqrt[Matrix[( {
{0, 1},
{-1, 0}
} )]] /. f_[Matrix[x__]] :> Matrix[MatrixFunction[f, x]]
</code></pre>
<p><code>Matrix</code> is an undefined symbol but I want to define some substitutions with it.</p>
| userrandrand | 86,543 | <p>As explained by @Mikado <code>Sqrt</code> is converted to <code>Power</code> which takes two arguments. One solution is to define:</p>
<pre><code>sqrt=Inactive[Sqrt]
</code></pre>
<p>and then use <code>sqrt</code> instead like</p>
<pre><code>sqrt[Matrix[({{0, 1}, {-1, 0}})]] /.
f_[Matrix[x__]] :> Matrix[MatrixF... |
3,173,349 | <p>The question:</p>
<p>Suppose G is a finite group of Isometries in the plane. Suppose p is fixed by G. Prove that G is conjugate into O(2).</p>
<p>What exactly does "conjugate into O(2)" mean and how would I proceed?</p>
| Will Jagy | 10,400 | <p>Lemma: if integers <span class="math-container">$a,b > 0$</span> and <span class="math-container">$ab = k^2$</span> and <span class="math-container">$\gcd(a,b) = 1,$</span> then both <span class="math-container">$a,b$</span> are squares. This is by unique factorization.</p>
<p>No need to consider negative <span ... |
3,173,349 | <p>The question:</p>
<p>Suppose G is a finite group of Isometries in the plane. Suppose p is fixed by G. Prove that G is conjugate into O(2).</p>
<p>What exactly does "conjugate into O(2)" mean and how would I proceed?</p>
| Ehsaan | 78,996 | <p>A positive integer <span class="math-container">$n$</span> is a square if and only if all of its prime factors occur with <em>even</em> multiplicity.</p>
<p>So this would be true of the integer <span class="math-container">$d:=n(n+1)(n+2)$</span> if it is a square. Since consecutive integers are coprime, and "twins... |
4,461,482 | <p>In <span class="math-container">$ΔABC$</span>, if <span class="math-container">$(a+b+c)(a−b+c)=3ac$</span>, then which of the following is <span class="math-container">$\color{green}{\text{True}}$</span>?</p>
<ul>
<li><span class="math-container">$\angle B=60^\circ $</span></li>
<li><span class="math-container">$\an... | insipidintegrator | 1,062,486 | <p><span class="math-container">$(a+b+c)(a-b+c)=3ac$</span>
<br> <span class="math-container">$(a+c)^2$</span> - <span class="math-container">$b^2$</span> = 3ac
<br> <span class="math-container">$\frac{a^2+c^2-b^2}{2ac}$</span> = <span class="math-container">$\frac{1}{2}$</span>
<br> Hence by the cosine rule for triang... |
255,902 | <p>I have a double lattice sum and I was wondering how I could calculate this with Mathematica. In particular, I have a function <span class="math-container">$F:\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$</span> which takes as arguments a pair of points <span class="math-container">$x,y\in\mathbb{R}^2$</span>, and... | kglr | 125 | <pre><code>sum1 = Total[F @@@ Tuples[Range[0, 3], {2, 2}]]
sum1 // Short
</code></pre>
<blockquote>
<pre><code>F[{0, 0}, {0, 0}] + F[{0, 0}, {0, 1}] + F[{0, 0}, {0, 2}] + << 250 >> +
F[{3, 3}, {3, 1}] + F[{3, 3}, {3, 2}] + F[{3, 3}, {3, 3}]
</code></pre>
</blockquote>
<pre><code>sum2 = Array[F[{#, #2}, {... |
3,817,760 | <p>In geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance point and center of mass of the triangle. Why is that the case?</p>
<p>This is the best explanation I could think of. I ho... | Kenneth Luo | 1,066,156 | <p>I have a fairly intuitive proof but not very rigorous. Firstly, the median of any triangle divides it into two triangles of equal area. This is because the median goes from one corner and bisects the opposing side, resulting in two triangles which have the same base since each is one half of the original side. They ... |
158,469 | <p>This is another exercise from Golan's book.</p>
<p><strong>Problem:</strong> Let $V$ be an inner product space over $\mathbb{C}$ and let $\alpha$ be an endomorphism of $V$ satisfying $\alpha^*=-\alpha$, where $\alpha^*$ denotes the adjoint. Show that every eigenvalue of $\alpha$ is purely imaginary.</p>
<p>My prop... | Potato | 18,240 | <p>Suppose $c$ is an eigenvalue of $\alpha$ and $v$ is a corresponding eigenvector. We have</p>
<p>$$c \langle v,v\rangle= \langle \alpha(v),v\rangle =\langle v, \alpha^*(v)\rangle =\langle v,-\alpha(v)\rangle=\langle v, -cv\rangle=\overline{\langle -cv, v\rangle} = \overline{-c\langle v,v\rangle}$$</p>
<p>Because $\... |
3,486,456 | <p>Suppose <span class="math-container">$f:X\rightarrow Y$</span> is a homeomorphism. Show that if <span class="math-container">$X$</span> is hausdorff then so is <span class="math-container">$Y$</span>.</p>
<p>My attempt:
Let <span class="math-container">$y_1,y_2\in Y$</span> be distinct, by bijectivity of <span clas... | Community | -1 | <p>Homeomorphisms preserve all topological properties, and the result follows. </p>
|
484,095 | <p>When taking the log of a matrix we have various choices, but fixing a particular choice, we should have</p>
<p>$$P^{-1}\log{(A)} P = \log(P^{-1}AP),$$</p>
<p>right? (Here $P \in GL$.)</p>
<p>It is supported by the notion that we can exponentiate both sides and it comes out true. Is there some snag here that I'm m... | Jim | 56,747 | <p>We have $b \equiv 1$ modulo $b - 1$ so for any $i$ we have $b^i \equiv 1$ modulo $b - 1$, but this <strong>doesn't</strong> come from "dividing" the summation, remember that
$$\sum x_iy_i$$
means
$$x_1y_1 + x_2y_2 + x_3y_3 + \cdots$$
so you can't divide out $\sum x_i$.</p>
<p>Anyway, $b^i \equiv 1$ modulo $b - 1$ i... |
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Zhanxiong | 192,408 | <p>Use the facts</p>
<blockquote>
<p>$$(a^x)' = a^x \ln a, \quad (\log_a(x))' = \frac{1}{x\ln a}$$
for $a > 0$ and $a \neq 1$.</p>
</blockquote>
<p>Then use the multiplication formula for differentiation to get
\begin{align}
& (10^x \log_{10}x)' \\
= & (10^x)'\log_{10}x + 10^x (\log_{10}x)' \\
= & ... |
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | zahbaz | 176,922 | <p>Your answer is correct, but you can simplify it further using the change of base formula. Where $\log x$ is in base $10$ and $\ln x$ is the natural log, you have</p>
<p>$$10^x\cdot \ln10\cdot \log x+\frac{1}{x\cdot \ln10}\cdot 10^x$$</p>
<p>$$=10^x\cdot \left(\ln10\cdot \color{green}{\log x}+\frac{1}{x\cdot \ln 10... |
971,333 | <p>Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. </p>
<p>I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to $y=e+1-x$ so $\int \limits_{y=\ln\left(x\right)}^{e+1-x}dy\int \:dx$, but I don't know how to find the limits of the ... | georg | 144,937 | <p>HINT:</p>
<p>y = ln(x), y = e+1-x</p>
<p>I would say, can guess the solution x = e, y = 1. Other real solution is not.</p>
|
17,285 | <p>I have problems in understanding few concepts of elementary set theory. I've choosen a couple of problems (from my problems set) which would help me understand this concepts. To be clear: it's not a homework, I'm just trying to understand elementary set's theory concepts by reading solutions.
<hr/>
<strong>Problem 1... | Arturo Magidin | 742 | <p><strong>Problem 1.</strong> Here's an example: take $A=\mathbb{N}$, $B=\{0,1,2,3\}$, and let $f\colon A\to B$ map the natural numbers to their remainder when divided by $4$ (so $f(3) = 3$, $f(15) = 3$, $f(21) = 1$, etc). Let $R$ be the equivalence relation on $B$ given by $aRb$ if and only if $a^2\equiv b^2 \pmod{4}... |
41,175 | <p>Suppose that I have two linear functions</p>
<pre><code>f[x_] := f0 + f1 x
g[x_] := g0 + g1 x
</code></pre>
<p>and a (possibly rather complicated) set of conditional expressions, obtained through Reduce. For example, we might have something like this:</p>
<pre><code>conditions = (f0 == f1 && g0 == 0) || ... | Kuba | 5,478 | <pre><code>Assuming[#, Simplify[{f[x], g[x]}]] & /@ List @@ conditions
</code></pre>
<blockquote>
<pre><code>{{f1 (1 + x), g1 x}, {g1 + g0 x, g0 + g1 x}}
</code></pre>
</blockquote>
<p>Which, technically but with switched constants, is what is desired.</p>
|
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | Heike | 46 | <p>Whit solution is using a <code>NestWhile</code> construction in combination with <code>Sow</code> and <code>Reap</code></p>
<pre><code>partitions[list_, {parts_List, offsets_List}] :=
Reap[
NestWhile[{RotateLeft[#[[1]]], RotateLeft[#[[2]]],
Sow[#[[3]][[;; #[[1, 1]]]]]; ArrayPad[#[[3]], {-#[[2, 1]], 0}]... |
2,540,640 | <p>Say I pick a random $n\times n$ matrix with entries uniformly chosen from $[0, 1]$.</p>
<p>a) What is the probability it is nilpotent?</p>
<p>b) What is the probability that, given it is nilpotent, it is similar to a given matrix?</p>
<p>To clarify part (b), there are finitely many possibilities for it when writt... | Robert Israel | 8,508 | <p>The nilpotent matrices form a variety of (I think) dimension $n^2-n$ in the $n \times n$ matrices (based on the fact that the coefficients of $\lambda^j$ for $j=0,\ldots,n-1$ in the characteristic polynomial must be $0$). You can then consider $n^2-n$-dimensional Hausdorff measure on this variety. Fix some nonzero v... |
2,540,640 | <p>Say I pick a random $n\times n$ matrix with entries uniformly chosen from $[0, 1]$.</p>
<p>a) What is the probability it is nilpotent?</p>
<p>b) What is the probability that, given it is nilpotent, it is similar to a given matrix?</p>
<p>To clarify part (b), there are finitely many possibilities for it when writt... | Community | -1 | <p>Let $N$ be the set of $n\times n$ complex nilpotent matrices. $N$ is an algebraic set but it is not a pure variety because its dimension is locally constant but not globally (except for eventual singularities). Let $A\in N$ and $U$ be its Jordan form. Note that the group $Gl_n$ acts as follows: </p>
<p>$(P,Z)\in GL... |
1,806,778 | <p>I'm currently studying for my logic exam, and looking into examples on DFA construction. </p>
<p>Assume the alphabet is {a, b}, and the language to be constructed is defined as follows: </p>
<pre><code>{w | w has at least three a's}
</code></pre>
<p>All models solutions I have come across look like the following:... | kennytm | 171 | <p>I'm not sure why there is a $(-2+\gamma+\ln 2)$, because $\left|\ln(x+1) - \Psi(x+\frac32)\right| < 0.0365$ itself is already satisfied.</p>
<p>The approximation comes from the <a href="https://en.wikipedia.org/wiki/Digamma_function#Computation_and_approximation" rel="nofollow">series expansion of $\exp\left(\Ps... |
1,806,778 | <p>I'm currently studying for my logic exam, and looking into examples on DFA construction. </p>
<p>Assume the alphabet is {a, b}, and the language to be constructed is defined as follows: </p>
<pre><code>{w | w has at least three a's}
</code></pre>
<p>All models solutions I have come across look like the following:... | gammatester | 61,216 | <p>I doubt your approximation. First I guess you missed a factor $2$ for $\ln 2$, the approximation should be
$$f(x) =\ln(x+1)-\Psi(x+3/2)-2+\gamma+2\ln 2 \cdot$$
If you omit the $2$ you have
$$f(0)= 0 -(2-\gamma-2\ln2) -2+\gamma+ \ln 2 = -4 + 2\gamma + 3\ln 2 \approx -0.766$$
while with the factor $2$ you have
$$f(0... |
297,251 | <p>In his well-known <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101110.pdf" rel="nofollow noreferrer">paper</a> Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:</p>
<p>We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \omega_1\}$ of m... | Joseph O'Rourke | 6,094 | <p>Not sure if this will help...</p>
<blockquote>
<p>Süß, Hendrik. "Fano Threefolds with 2-Torus Action: A Picture Book." <em>Documenta Mathematica</em> 19 (2014): 905-940.
(<a href="https://www.math.uni-bielefeld.de/documenta/vol-19/30.pdf" rel="nofollow noreferrer">PDF download</a>.)
<br />
<strong>Abstract<... |
2,529,088 | <p>Well, I'm learning olympiad inequalities, and most of the books go like this:</p>
<blockquote>
<p>Memorize holder-cauchy-jensen-muirhead, then just dumbass (i.e bash and homogenize) and use the above theorems.</p>
</blockquote>
<p>Needless to say, I feel extremely disintersted in inequalities for this approach. ... | Jaideep Khare | 421,580 | <p>$$\underbrace{(2^n)}_{\text{even}}\underbrace{( 2^{m-n}-1)}_{\text{odd}}=112=\underbrace{2^4}_{\text{even}}\times \underbrace{7}_{\text{odd}}$$</p>
<p>Thus, $$2^n=2^4 \implies \color{blue}{n=4}$$ and $$2^{m-4}-1=7 \implies 2^{m-4}=2^3 \implies \color{blue}{m=7}$$</p>
|
2,529,088 | <p>Well, I'm learning olympiad inequalities, and most of the books go like this:</p>
<blockquote>
<p>Memorize holder-cauchy-jensen-muirhead, then just dumbass (i.e bash and homogenize) and use the above theorems.</p>
</blockquote>
<p>Needless to say, I feel extremely disintersted in inequalities for this approach. ... | Robert Z | 299,698 | <p>Hint. Note that $m>n\geq 1$ and
$$2^n\cdot (2^{m-n} -1)=112=2^4\cdot 7.$$</p>
|
4,290,960 | <p>In machine learning (and not only), it is very common to see concatenation of different feature vectors into a single one of higher dimension which is then processed by some function. For example, feature vectors computed for an image at different scales are concatenated to form a multi-scale feature vector which is... | Will Jagy | 10,400 | <p>let me make this Community Wiki, people sometimes dislike computed lists; crucial, however, in exploring diophantine equations, looking for patterns.</p>
<p>Alright, there are always examples with repeats; for any positive <span class="math-container">$b,$</span> the triple with <span class="math-container">$a=1,... |
3,219,187 | <p>The cancellation law for the multiplication of natural numbers is:</p>
<p><span class="math-container">$$\forall m, n\in\mathbb N, \forall p\in\mathbb N-\{0\}, m\cdot p=n\cdot p\Rightarrow m=n.$$</span></p>
<p>Is it possible to show this using induction?</p>
<p>I tried to define <span class="math-container">$$X=\... | Mauro ALLEGRANZA | 108,274 | <p><em>Hint</em></p>
<p>You have : <span class="math-container">$m⋅p=(n+1)⋅p$</span>.</p>
<p>But you know that <span class="math-container">$(n+1) \ne 0$</span> and the assumption is that <span class="math-container">$p \ne 0$</span>. Thus <span class="math-container">$m⋅p=(n+1)⋅p \ne 0$</span> and from it and <span ... |
180,121 | <p>Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\Psi_p:\mathcal{F}_p \rightarrow f_{\ast}f^{\ast}(\mathcal{F})_p=f_{p,\ast}f_p^{\ast}(\mathcal{F}_p)$$</p>
<p>I'm inter... | Benjamin Schmidt | 14,385 | <p>The bounded derived category of coherent sheaves is a nice technical tool to understand this question more in depth. Assume that that $A$ and $B$ are bounded complexes of coherent sheaves. Then by using derived functors you get a general version of the projection formula
$$Rf_*(B) \otimes^L A \cong Rf_*(B \otimes^L ... |
9,880 | <p>The ban on edits that change only a few characters can be quite annoying in some cases. I don't have a problem with trying to keep trivial spelling corrections and such out of the peer review queue, but sometimes a small edit is semantically significant. For example, it makes perfect sense to edit an otherwise good ... | Carl Mummert | 630 | <p>There is no absolute ban on small edits. It's up to each person who has enough rep to edit to decide whether a change should be made. </p>
<p>The main reason is to avoid small changes is that an edit can bump a question onto the front page. This can be particularly problematic if someone edits a few dozen questions... |
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Aldo Guzmán Sáenz | 92,139 | <p>About showing that $g(1)=1$: If we assume that $g(0)=0$ and $g(2)\neq 0$, we can prove that $g(1)=1$ as follows:</p>
<p>We take $x=y=2$, and substitute:</p>
<p>$g(2+2)+g(2)g(2)=g(2\cdot 2)+g(2)+g(2)$</p>
<p>thus</p>
<p>$g(4)+g(2)^2=g(4)+2g(2)$, and since we are assuming that $g(2)\neq 0$, then</p>
<p>$g(2)=2$.<... |
2,086,598 | <p>Let's start with a well known limit:
$$\lim_{n \to \infty} \left(1 + {1\over n}\right)^n=e$$</p>
<p>As $n$ approaches infinity, the expression evaluates Euler's number $e\approx 2.7182$. Why that number? What properties does the limit have that leads it to 2.71828$\dots$ and why is this number important? Where can ... | MoebiusCorzer | 283,812 | <p>As Walter Rudin puts it in his famous "Real and Complex Analysis":</p>
<blockquote>
<p>This is the most important function in mathematics. It is defined, for every complex number <span class="math-container">$z$</span>, by the formula</p>
<p><span class="math-container">$$\exp(z)=\sum_{n=0}^{\infty}\frac{z... |
3,663,267 | <p>in a linear algebra class i was given a theorem: If <span class="math-container">$\left\{\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{n}\right\}$</span> is a linearly independent subset of <span class="math-container">$\mathbb{R}^{n}$</span> then
<span class="math-container">$
\mathbb{R}^{n}=\oper... | Dabouliplop | 426,049 | <p>What you are trying to do is to understand why if <span class="math-container">$i$</span>, <span class="math-container">$j$</span>, <span class="math-container">$k$</span> are three distinct variables (as in programming languages), the unique substitution <span class="math-container">$j \leftarrow j + abi$</span> ha... |
2,796,854 | <p>what happens if both the second and first derivatives at a certain point are $0$? Is it an infliction point, an extremum point, or neither? Can we say anything at all about a point in such a case? </p>
| Surb | 154,545 | <p>You can't say anything. Example : $$x\longmapsto x^3\quad x\longmapsto x^4\quad \text{and}\quad x\longmapsto\begin{cases} x^3\sin\left(\frac{1}{x}\right)&x\neq 0\\ 0&x=0\end{cases}.$$</p>
|
1,154,690 | <p>We have instituted random drug testing at our company. I was charged with writing the code to generate the weekly random list of employees. We've gotten some complaints because of people getting picked more frequently than they expected and our lawyer wants some evidence that these events fall within the bell curve ... | TravisJ | 212,738 | <p>Let $n$ be the number of weeks you've sampled. The for any individual, the probability that they are chosen on any given week is $p=.02$. Then, the probability, that after $n$ weeks any individual has been chosen at least 3 times is: </p>
<p>\begin{align*} \mathbb{P}(\text{chosen } \geq 3 \text{ times} )
&= ... |
3,407,174 | <p>I'm dealing with a specific polynomial function. The first derivative of it is displayed below. As you can see, it has the shape of an asymetric function. But what does this tells us about the initial function in general and in terms of finding a maximum or a minimum for it? I know it is a general question, but I fi... | J.G | 293,121 | <p>Note that <span class="math-container">$ee^T$</span> is rank-one with a single nonzero eigenvalue of <span class="math-container">$e$</span> (with eigenvector <span class="math-container">$m$</span>). So <span class="math-container">$(1+1/\gamma)I-ee^T/m$</span> has eigenvalues <span class="math-container">$1+1/\gam... |
988,581 | <p>Our teacher challenge us a question and it goes like this:
The derivative of a function is define such as </p>
<p>$$\begin{cases} 1 & \text{if } x>0 \\ 2 & \text{if } x=0 \\ -1 &\text{if } x<0\end{cases} $$</p>
<p>Now he said that "$2$, if $x=0$" is not possible.He asked us to proved that the d... | Lucian | 93,448 | <ul>
<li><p>On one hand, the series is strictly increasing.</p></li>
<li><p>On the other hand, <span class="math-container">$n^3+2\le2n^3$</span> and <span class="math-container">$n^4+3n^2+1\ge n^4$</span>, implying <span class="math-container">$t_n\le\ldots~\Big($</span>I'll let you fill in the dots, and draw the conc... |
381,309 | <p>Let <span class="math-container">$T\in B(\mathcal{H} \otimes \mathcal{H})$</span> where <span class="math-container">$\mathcal{H}$</span> is a Hilbert space. We can define operators
<span class="math-container">$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$</span>
and if <span class="math-container">$\Sigma: ... | Gerald Edgar | 454 | <p>Not an answer (does not use translation invariance)</p>
<p>Another non-measurable set, which may generalize more easily, is the <a href="https://math.stackexchange.com/questions/169714/whats-application-of-bernstein-set">Bernstein set</a> ... That is, a set <span class="math-container">$E$</span> such that for every... |
3,777,049 | <p>Suppose that <span class="math-container">$(H_i)_{i \in I}$</span> is a collection of closed <strong>orthogonal</strong> subspaces of the Hilbert space <span class="math-container">$H$</span>. Suppose that <span class="math-container">$\sum_{i \in I} \Vert x_i \Vert^2 < \infty$</span>. Prove that <span class="mat... | Kavi Rama Murthy | 142,385 | <p>Some mistakes have already been pointed out. So I will give a valid proof.</p>
<p><span class="math-container">$ \sum \|x_i\|^{2} <\infty$</span> implies that <span class="math-container">$x_i=0$</span> for al but countably many <span class="math-container">$i$</span>. Hence the result reduces to the case of co... |
3,777,049 | <p>Suppose that <span class="math-container">$(H_i)_{i \in I}$</span> is a collection of closed <strong>orthogonal</strong> subspaces of the Hilbert space <span class="math-container">$H$</span>. Suppose that <span class="math-container">$\sum_{i \in I} \Vert x_i \Vert^2 < \infty$</span>. Prove that <span class="mat... | Danny Pak-Keung Chan | 374,270 | <p>Firstly, let us clarify the meaning of the symbol <span class="math-container">$\sum_{i\in I}x_{i}$</span>.
Let <span class="math-container">$\mathcal{C}$</span> be the collection of all finite subsets of <span class="math-container">$I$</span>.
Then <span class="math-container">$(\mathcal{C},\subseteq)$</span> is a... |
3,745,097 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<p>If <span class="math-container">$F$</span> is a non-empty countable subset of <span class="math-container">$\mathbb R$</span>, prove that <span class="math-container">$F$</span> is not an open set, but that <span class="math-contain... | Ross Millikan | 1,827 | <p>Finite unions of closed sets are closed, but infinite unions need not be. Look for a set of points that approaches a limit. If that limit is not in the set, the set is not closed.</p>
|
3,962,058 | <p>Let <span class="math-container">$A$</span> be a commutative ring with identity, and <span class="math-container">$P$</span> a prime ideal of <span class="math-container">$A$</span>. I want to show that <span class="math-container">$\lim_{f \notin P} A_{f} = A_{P}$</span>. <span class="math-container">$A_f$</span> i... | Joshua P. Swanson | 86,777 | <p>It seems to me you're assuming the direct limit exists, but justifying that requires some sort of explicit construction anyway, which you're trying to avoid.</p>
<p>I think what you really what you want to do is to show that <span class="math-container">$A_P$</span> (with the system of natural maps <span class="math... |
1,527,675 | <p>I'm trying to proof that the subset $H$ of $G$ together with $\circ$ is also a group i.e. a subgroup and I'm stuck in the proof that $(H,\circ)$ is associative. </p>
| Paul Orland | 42,566 | <p>Let $a$, $b$, and $c$ be in $H$. Then since $H$ is a subset of $G$, we know $a$, $b$, and $c$ are in $G$ and satisfy the associative law: $a\circ(b\circ c) = (a\circ b)\circ c$. Therefore $\circ$ is associative for $H$.</p>
<p>Now, not every subset $H$ of $G$ is a subgroup, but your question seems to imply you'... |
1,527,675 | <p>I'm trying to proof that the subset $H$ of $G$ together with $\circ$ is also a group i.e. a subgroup and I'm stuck in the proof that $(H,\circ)$ is associative. </p>
| lhf | 589 | <p>By definition, a subset $H$ of a group $G$ is a subgroup when $H$ is a group using the operation of $G$.</p>
<p>In principle, this would require you to check all group properties for $H$:
$H$ is closed under the operation, there is a neutral element, there exists an inverse for each element in $H$, and that the ope... |
2,010,329 | <p>The function under consideration is:</p>
<p>$$y = \int_{x^2}^{\sin x}\cos(t^2)\mathrm d t$$</p>
<p>Question asks to find the derivative of the following function. I let $u=\sin(x)$ and then $\tfrac{\mathrm d u}{\mathrm d x}=\cos(x)$. Solved accordingly but only to get the answer as
$$ \cos(x)\cos(\sin^2(x))-\cos(... | layman | 131,740 | <p>Notice that \begin{split} \int \limits_{x^{2}}^{\sin{x}} \cos{t^{2}}\,dt &= \int \limits_{x^{2}}^{a} \cos{t^{2}}\,dt + \int \limits_{a}^{\sin{x}} \cos{t^{2}}\,dt \\ &= \underbrace{-\int \limits_{a}^{x^{2}} \cos{t^{2}}\,dt}_{A} + \underbrace{\int \limits_{a}^{\sin{x}} \cos{t^{2}}\,dt}_{B} \end{split}</p>
<p... |
2,724,866 | <p>This question caught my eye from "How to integrate it" by Sean M. Stewart. I have attempted it for fun and am now stuck. </p>
<p><strong>Question</strong></p>
<p>Let $f$ and $g$ be continuous bounded functions on some interval $[a,b]$ such that $\int_a^b|f(x)-g(x)|dx=0$. Show that $\int_a^b|f(x)-g(x)|^2dx=0$.<... | Ingix | 393,096 | <p>Are you sure that the condition is for $f$ and $g$ to be continuous? In that case, if $f$ and $g$ differ at one point $x_0$, then $f-g$ has the same sign and $|f-g| > \epsilon > 0 $ in some small interval around $x_0$, so the integral can't be zero. In other words, under that condition we get the much better r... |
2,724,866 | <p>This question caught my eye from "How to integrate it" by Sean M. Stewart. I have attempted it for fun and am now stuck. </p>
<p><strong>Question</strong></p>
<p>Let $f$ and $g$ be continuous bounded functions on some interval $[a,b]$ such that $\int_a^b|f(x)-g(x)|dx=0$. Show that $\int_a^b|f(x)-g(x)|^2dx=0$.<... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Replace $f- g$ with $h$. It suffices to show that
$$\int_a^b |h|=0 \implies \int_a^b |h|^2=0 \quad \forall\, h \in C([a,b])$$
As @KingTut points out, this follows trivially from
$$\forall\, h \in C([a,b]), \int_a^b |h|=0 \implies h \equiv 0.$$
Its contrapositive form has a straightforward proof: let $h(x_0) \ne 0$ ... |
4,180,344 | <p>I started learning from Algebra by Serge Lang. In page 5, he presented an equation<br><i></p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a commutative monoid, and <span class="math-container">$x_1,\ldots,x_n$</span> elements of <span class="math-container">$G$</span>. Let <span class="math-contai... | Bernard | 202,857 | <p><span class="math-container">$\psi(\nu)$</span> is not a mapping, but the value of the map <span class="math-container">$\psi $</span> at <span class="math-container">$\nu$</span>. In other words, the <span class="math-container">$n$</span>-uple <span class="math-container">$(x_{\psi(1)},x_{\psi(2)},\dots,x_{\psi(n... |
189,308 | <p>I have this problem: Find integration limits and compute the following integral.</p>
<p>$$\iiint_D(1-z)\,dx\,dy\,dz \\ D = \{\;(x, y, z) \in R^3\;\ |\;\; x^2 + y^2 + z^2 \le a^2, z\gt0\;\}$$</p>
<p>I can compute this as an indefinite integral but finding integration limits beats me. As indefinite integral the resu... | Santosh Linkha | 2,199 | <p>Change the limits as
$$ 8*\int_0^a \int_0^{\sqrt{a^2 - x^2}} \int_0^{\sqrt{a^2 - x^2 - y^2}} (1 - z) \,dz \,dy \,dx $$
This integral should give you the volume of sphere. Note that it is easy to remember the limits.
$$ 8*\int_0^a \int_0^{\sqrt{a^2 - x^2}} \int_0^{\sqrt{a^2 - x^2 - y^2}} 1 \cdot \,dz \,dy \,dx = {4... |
3,379,756 | <p>Why does the close form of the summation
<span class="math-container">$$\sum_{i=0}^{n-1} 1$$</span> equals <span class="math-container">$n$</span> instead of <span class="math-container">$n-1$</span>?</p>
| user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\sum_{i=0}^{n-1}= \overbrace{{1+1+1+\cdots+1}}^\color{red}{\text{n terms from 0 to n-1}}=n$$</span></p>
|
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| José Carlos Santos | 446,262 | <p>If <span class="math-container">$n<k$</span>, then<span class="math-container">\begin{align}\frac{2^{2\times3^n}}{2^{2\times3^k}}&=4^{3^n-3^k}\\&\leqslant4^{3^{k-1}-3^k}\\&=4^{-2\times3^{k-1}}\\&\leqslant4^{-(k-1)},\end{align}</span>because <span class="math-container">$2\times3^{k-1}\geqslant k-1... |
3,407,474 | <p>I can understand that The set <span class="math-container">$M_2(2\mathbb{Z})$</span> of <span class="math-container">$2 \times 2$</span>
matrices with even integer entries is an infinite non commutative ring. But why It doesn't have any unity? I think <span class="math-container">$I(2\times2)$</span> is a unity for ... | John Hughes | 114,036 | <p>The <span class="math-container">$2\times 2$</span> identity has entries <span class="math-container">$1,0,0,1$</span>. Are all of those are even numbers? </p>
|
3,407,474 | <p>I can understand that The set <span class="math-container">$M_2(2\mathbb{Z})$</span> of <span class="math-container">$2 \times 2$</span>
matrices with even integer entries is an infinite non commutative ring. But why It doesn't have any unity? I think <span class="math-container">$I(2\times2)$</span> is a unity for ... | Chris Leary | 2,933 | <p>If your matrix ring had an identity element <span class="math-container">$I,$</span> then the square of its determinant would have to be <span class="math-container">$1$</span> (or <span class="math-container">$0$</span>), since <span class="math-container">$I^2=I.$</span> The determinant cannot be <span class="math... |
1,424,297 | <p>Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?</p>
| André Nicolas | 6,312 | <p>Multiply top and (missing) bottom by $1+\cos(1/n)$. After using the identity $1-\cos^2 t=\sin^2 t$, we get that we want
$$\lim_{n\to\infty}\left(\frac{1}{1+\cos(1/n)}\cdot \frac{\sin^2(1/n)}{(1/n)^2}\right).$$
The rest should follow from a limit you know. </p>
|
1,424,297 | <p>Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?</p>
| Hagen von Eitzen | 39,174 | <p>Using $$\cos x-\cos y =2\sin\frac{x+y}{2}\sin\frac{y-x}{2}$$
we find with $x=0$ and $y=\frac1n$
$$ 1-\cos\frac1n=2\sin^2\frac1{2n}$$
Now if you know $\lim_{x\to 0}\frac{\sin x}x=1$, you also get $\lim_{n\to\infty}2n\sin\frac1{2n}=1$ so ultimately
$$\lim_{n\to\infty} n^2\left(1-\cos \frac1n\right) =\lim_{n\to\infty}2... |
1,424,297 | <p>Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?</p>
| Bernard | 202,857 | <p><em>With equivalents:</em></p>
<p>It's a classic fact that $\;1-\cos\dfrac1n\sim_\infty\dfrac1{2n^2}$ (for a proof see Taylor development at order $2$ of $\cos$), hence $$n^2\Bigl(1-\cos \frac1n\Bigr)\sim_\infty\frac{n^2}{2n^2}=\frac12.$$</p>
|
2,597,126 | <p>Given the following problem:</p>
<blockquote>
<p>Applying the division algorithm, prove that all whole numbers that are at the same time a square and a cube have the form $7k$ or $7k+1$.</p>
</blockquote>
<p>I am unable to interpret what it is asking of me and therefore I am unable to provide any solutions. Coul... | Dr. Sonnhard Graubner | 175,066 | <p>we have $$x\equiv 0,1,2,3,4,5,6\mod 7$$ then
$$x^6\equiv 0,1,1,1,1,1,1 \mod 7$$ you should check this</p>
|
3,571,637 | <p>I was reading Linear Algebra by Hoffman Kunze, and encountered this in the Theorem 8 of the Chapter Coordinates, the theorem is stated below : </p>
<p><a href="https://i.stack.imgur.com/6xFlJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6xFlJ.png" alt="coo"></a></p>
<p>What I get about is in ... | justadzr | 672,528 | <p><strong>Theorem 8</strong>: this theorem actually starts with the set <span class="math-container">$\mathcal B'$</span> which is supposed to be our aimed basis. Therefore, Kunze says: let's define a set of vectors <span class="math-container">$\mathcal B'$</span> where its elements are defined by
<span class="math-... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Jack M | 30,481 | <p>The <a href="http://en.wikipedia.org/wiki/Gauss-Wantzel_theorem" rel="noreferrer">Gauss-Wantzel theorem</a> on constructible polygons immediately springs to mind. This states that a regular $n$-gon is constructible with a straightedge and compass iff $n$ is the product of a power of $2$ and a collection of distinct ... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | chris williams | 219,521 | <p>Construction of polygons that have prime number properties are more complex to construct than composite numbers, a polygon with composite properties can easily constructed by splitting each compostite fraction into half,
example, circle in half = 2
Circle in quarters = 4
Continue Quarters in half = 8 (notice 6 is m... |
838,631 | <p>This question may be far too easy for this site but I always seem to get stuck when it comes to the triangle inequality.</p>
<p>For example, I am trying to prove that differentiability implies Lipschitz continuity. Ok, I don't really have a hard time with this except for the step when using the triangle inequality.... | Surb | 154,545 | <p>Seems that the <a href="http://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality" rel="nofollow">reverse triangle inequality</a> is used in your case: $| \ |a|-|b|\ | \leq |a-b|$ for any real numbers $a,b$.</p>
<p>Applying this in the setting of your problem:</p>
<p>$$ |f(x)-f(x_0)|-|f'(x_0)(x-... |
1,981,553 | <p>When a person asks: "What is the smallest number (natural numbers) with two digits?"</p>
<p>You answer: "10".</p>
<p>But by which convention 04 is no valid 2 digit number?</p>
<p>Thanks alot in advance</p>
| Soham | 242,402 | <p>Simple answer would be $0$ has no value infront of a number.So,$04,004,0004,000000000004$ are all same.As,this $0$ has no meaning or significance in the representation of the number,$0$ infront of a number is not considered.</p>
|
369,104 | <p>How do I take an algebraic expression and construct a tree out of it?</p>
<p>Sample equation:</p>
<p>((2 + x) - (x * 3)) - ((x - 2) * (3 + y))</p>
<p>If somebody can teach me in steps, that would be really helpful!</p>
| vonbrand | 43,946 | <p>You are looking for a <a href="http://en.wikipedia.org/wiki/Abstract_syntax_tree" rel="nofollow">syntax tree</a>, and that is part of what <a href="http://en.wikipedia.org/wiki/Parsing" rel="nofollow">parsing</a> is all about. This is more suited for <a href="http://cs.stackexchange.com">http://cs.stackexchange.com<... |
3,868,523 | <blockquote>
<p>Given the bases <em>a</em> = {(0,2),(2,1)} and <em>b</em> = {(1,0),(1,1)} compute the change of coordinate matrix from basis <em>a</em> to <em>b</em>.<br />
Then, given the coordinates of <em>z</em> with respect to the basis <em>a</em> as (2,2), use the previous question to compute the coordinates of <e... | Community | -1 | <p>You need to change the coordinates from the basis <span class="math-container">$a$</span> to the canonical basis (multiplying by <span class="math-container">$A$</span>), then change from the canonical basis to the base <span class="math-container">$b$</span> (multiplying by <span class="math-container">$B^{-1}$</sp... |
1,470,378 | <p>I am not sure i formatted it right but I was studying for calculus and came across a problem I couldn't compute. </p>
<p>$$\int \frac{x^2}{x^2-2}dx$$</p>
<p>I have not learned partial fractions yet so if this is a case where that is used, the techniques might not work for me. </p>
<p>What I have tried. to do i... | Aditya | 278,230 | <p>Use partial fraction method.
Add and subtract 2 in the numerator and then the integration boggs down tona simple integration.</p>
|
1,299,630 | <p>I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$).</p>
<p>I immediately see a problem with <em>e.g.</em> Cauchy's theorem (there's no $\pi$ available) and perhaps even defining a "analytical function" co... | Fallen Apart | 201,873 | <p>Given a continuous mapping $F:X\rightarrow Y$ it induces continuous map $F^{**}:X^{**}\rightarrow Y^{**}$ given by the formula
$$[F^{**}(x^{**})](y^*):=x^{**}(F^*(y^*)).$$
This assigment is functorial, i.e. it has two properties:</p>
<p>$(id_X)^{**}=id_{X^{**}}$ and $(G\circ F)^{**}=G^{**}\circ F^{**}.$</p>
<p>Add... |
4,117,377 | <p>In calculating the integral</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx$$</span></p>
<p>by contour integration, we use</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx = \int_{-\infty}^{\infty} \frac{\operatorname{Im}(e^{ix})}{x}dx =\operatorname{I... | Thusle Gadelankz | 840,795 | <p>In order to say that the Cauchy principal value agrees with the ordinary value of an improper integral you only need to know that the integral coverges, which you may show using a convergence test. This is because the Cauchy principal value is simply a particularly easy way of taking the limit. The Cauchy principal ... |
4,117,377 | <p>In calculating the integral</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx$$</span></p>
<p>by contour integration, we use</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx = \int_{-\infty}^{\infty} \frac{\operatorname{Im}(e^{ix})}{x}dx =\operatorname{I... | Community | -1 | <p>Since this is a question about rigor we should state what integral we are using. From (1) This integral is not Lebesgue integrable as it is not absolutely convergent, it is an improper Riemann integral though so we will use the Riemann integral.</p>
<p>Let us have a visual of the function being integrated</p>
<p><a ... |
4,117,377 | <p>In calculating the integral</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx$$</span></p>
<p>by contour integration, we use</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx = \int_{-\infty}^{\infty} \frac{\operatorname{Im}(e^{ix})}{x}dx =\operatorname{I... | robjohn | 13,854 | <p><strong>Why the Cauchy Principal Value Gives the Proper Value</strong></p>
<p>Your question boils down to the following
<span class="math-container">$$
\begin{align}
&\mathrm{Im}\left(\mathrm{PV}\int_{-\infty}^\infty\frac{e^{ix}}x\,\mathrm{d}x\right)\\
&=\mathrm{Im}\left(\mathrm{PV}\int_{-\infty}^\infty\frac... |
1,075,505 | <p>I have the following function:</p>
<p>$$f(x+iy) = x^2+iy^2$$</p>
<p>My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we use to check where a function is differentiable?</p>
<p>I know the Cauchy-Riemann equations, and that $... | Harriet Grigg | 390,879 | <p>A perhaps more accessible perspective is to understand that the definition of the complex derivative, as in the real case, relies on the existence and uniqueness of a certain limit. In particular, given a point and a function, one considers evaluation of the function in a small neighbourhood of the point in question... |
2,234,737 | <p>$f:\mathbb{R}^m\to\mathbb{R}$ differentiable such that $f(x/2) = f(x)/2$, show $f$ is linear.</p>
<p>I tried to do:</p>
<p>$f(a+b) = f(2(a+b)/2) = f(2(a+b))/2$</p>
<p>but that wouldn't turn into $f(a)+f(b)$</p>
<p>In the same way: $f(a)/2 + f(b)/2 = f((a+b)/2)$ which won't help.</p>
<p>Also, I need to use the d... | Breno Pinheiro | 740,066 | <p>Let <span class="math-container">$f:\mathbb{R}^n\rightarrow \mathbb{R}$</span> be a differentiable function, then f linear <span class="math-container">$\Leftrightarrow$</span> <span class="math-container">$\dfrac{\partial f(a)}{\partial v}=f(v)$</span></p>
<p><span class="math-container">$\Rightarrow$</span> <span... |
567,668 | <p>In my analysis class we are discussing integrable functions and I need help thinking through something. I don't want any answers, just what should I be looking at? (hints)</p>
<p><img src="https://i.stack.imgur.com/qKte6.png" alt="enter image description here"></p>
| Berci | 41,488 | <p><strong>Hints:</strong> If $N$ is a normal subgroup of $G_1\times G_2$, as the projections $p_i:G_1\times G_2\to G_i$ are surjective, they map $N$ onto a normal subgroup.</p>
<p>For the case both $p_i(N)=G_i$, let $M:=\{g_1\mid (g_1,1)\in N \}$. Then $M$ is again a normal subgroup (of $G_1$). If $M=G_1$ we are read... |
1,522,062 | <p>Identify for which values of $x$ there is subtraction of nearly equal numbers, and find an alternate form that avoids the problem:
$$E = \frac{1}{1+x} - \frac{1}{1-x} = -\frac{2x}{1-x^2} = \frac{2x}{x^2-1} $$
How come $-2x/(1-x^2)$ can be changed to $2x(x^2 - 1)$ according to the homework solutions? Why does the den... | team-erdos | 288,344 | <p>In general, for numbers $a,b$ and $c$, we may write
$$
\frac a{b-c}=\frac a{-1(c-b)}=\frac 1{-1} \cdot \frac a{c-b}=\frac {-1}{1} \cdot \frac a{c-b}=\frac{-a}{c-b}.
$$</p>
<p>In your case, we have $a=-2x$, $b=1$, and $c=x^2$. So,
$$
\frac{-2x}{1-x^2}=\frac{-2x}{-1(x^2-1)}=\frac 1{-1}\cdot \frac{-2x}{x^2-1}=\frac {-... |
1,686,904 | <blockquote>
<p>How can we prove that <span class="math-container">$$\binom{n}{1}\binom{n}{2}^2\binom{n}{3}^3\cdots \binom{n}{n}^n \leq \left(\frac{2^n}{n+1}\right)^{\binom{n+1}{2}}$$</span></p>
</blockquote>
<p><span class="math-container">$\bf{My\; Try::}$</span> Using <span class="math-container">$\bf{A.M\geq G.M\;,... | DeepSea | 101,504 | <p><strong>Hint</strong>: You are close to the answer. Here is the right way: Apply the AM-GM:</p>
<p>$(a_1+2a_2+\cdots + na_n)^{\binom{n+1}{2}} \geq \binom{n+1}{2}^{\binom{n+1}{2}}a_1a_2^2\cdots a_n^n$, with $a_k = \binom{n}{k}$, and the left side is a popular sum that can be calculated by several methods one of whic... |
244,489 | <p>Given:</p>
<p>${AA}\times{BC}=BDDB$</p>
<p>Find $BDDB$:</p>
<ol>
<li>$1221$</li>
<li>$3663$</li>
<li>$4884$</li>
<li>$2112$</li>
</ol>
<p>The way I solved it:</p>
<p>First step - expansion & dividing by constant ($11$):
$AA\times{BC}$=$11A\times{BC}$</p>
<ol>
<li>$1221$ => $1221\div11$ => $111$</li>
<li>$3... | Patrick M | 48,933 | <p>So, assume that $0 \leq a \leq 1$ and $0\leq b\leq 1$. I'll address the boundary conditions later.[1] You have square A, inside of a unit square. When the square A is touching a corner, you have the most space available for square B.[2] Assuming that the square is touching a corner, you are left with two rectang... |
3,364,317 | <p>Points A and B fixed, and point C moves on circle such that ABC acute triangle. <span class="math-container">$AT = BT$</span> and <span class="math-container">$TM \perp AC, \, TN \perp BC$</span>. How can I proove that all the middle perpendiculars (perpendicular bissector) to <span class="math-container">$MN$</span... | sirous | 346,566 | <p>Hint:Let's mark perpendicular bisector as (PB).
If C is coincident on A or B then M and N will be coincident on A and B respectively and the (PB) of MN is exactly the (PB) of AB.Also when the triangle is isosceles MN is parallel with AB, so again (PB) of MN and AB are coincident. That is the point is on the (PB) of... |
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| Chinny84 | 92,628 | <p>$$
(x^2-1)^n = \sum_{k=0}^n\left(\matrix{n\\k}\right)(-1)^{n-k}(x^2)^k
$$
or your problem becomes
$$
\frac{d^n}{dx^n}\sum_{k=0}^n\left(\matrix{n\\k}\right)(-1)^{n-k}(x^2)^k = \sum_{k=0}^n\left(\matrix{n\\k}\right)(-1)^{n-k}\frac{d^n}{dx^n}x^{2k}
$$
Can you take it from here?</p>
|
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| Rhys Hughes | 487,658 | <p>We have:
$$y=(x^2-1)^n$$
Using the chain rule: $$y=a[f(x)]^n\to\frac{dy}{dx}=an[f'(x)][f(x)]^{n-1}$$
We get:
$$\frac{dy}{dx}=2xn(x^2-1)^{n-1}$$
Then:
$$\frac{d^2y}{dx^2}=2xn\cdot2x(n-1)\cdot(x^2-1)^{n-2}=n(n-1)(2x)^2(x^2-1)^{n-2}$$
$$\frac{d^3y}{dx^3}=2xn\cdot2x(n-1)\cdot2x(n-2)\cdot(x^2-1)^{n-2}=(2x)^3[n(n-1)(n-2)]... |
1,060,213 | <p>I have searched the site quickly and have not come across this exact problem. I have noticed that a Pythagorean triple <code>(a,b,c)</code> where <code>c</code> is the hypotenuse and <code>a</code> is prime, is always of the form <code>(a,b,b+1)</code>: The hypotenuse is one more than the non-prime side. Why is this... | poetasis | 546,655 | <p>The subset of triples where the GCD of A,B,C is an odd square and where <span class="math-container">$C-B=(2n-1)^2$</span> is also a group of distinct sets of triples as shown below.</p>
<p><span class="math-container">$$\begin{array}{c|c|c|c|c|}
\text{$Set_n$}& \text{$Triple_1$} & \text{$Triple_2$} &... |
3,554,555 | <p>Let <span class="math-container">$S_n=1-2^{-n}$</span>. Prove that the sequence <span class="math-container">$S_n$</span> converges to <span class="math-container">$1$</span> as <span class="math-container">$n$</span> approaches infinity. </p>
<p>I let <span class="math-container">$N = \lceil\log_2 \varepsilon\rcei... | P. Lawrence | 545,558 | <p>Expansion of <span class="math-container">$2^n=(1+1)^n$</span> by the binomial theorem shows that <span class="math-container">$2^n>n.$</span> Thus <span class="math-container">$2^{-n}<\frac {1}{n}$</span> so <span class="math-container">$$|1-(1-2^{-n})|=2^{-n}< \frac {1}{n}$$</span>. You can take it from t... |
1,115,389 | <p>Given: </p>
<p>$$\sum_{n = 0}^{\infty} a_nx^n = f(x)$$</p>
<p>where:</p>
<p>$$a_{n+2} = a_{n+1} - \frac{1}{4}a_n$$</p>
<p>is the recurrence relationship for $a_2$ and above ($a_0$ and $a_1$ are also given).</p>
<p>Is there a nice closed form to this pretty recurrence relationship?</p>
| Alex R. | 22,064 | <p>If you think about it, placing 9 rooks on a 9x9 chessboard so that they don't attack each other is equivalent to asking how many permutations there are of the numbers $1,2,\cdots,9$. This is because each allowed rook configuration corresponds to a $9\times 9$ permutation matrix $M$, with $M_{ij}=1$ if there's a rook... |
3,677,303 | <p>Is there an example of topological space being both sequentially compact and Lindelof but noncompact?</p>
| nessy | 785,381 | <p>No.</p>
<p>Let <span class="math-container">$X$</span> be a sequentially compact and Lindelöf topological space. Then <span class="math-container">$X$</span> is countably compact, since it is sequentially compact. (*) It is by definition that countably compact and Lindelöf space is compact.</p>
<p>(*) Countably c... |
801,574 | <p>Let $A$ be a complex $n\times m$ matrix ($n<m$), which is row full rank. For any $m\times m$ matrix $M$, such that $AM$ is still row full rank, can we find an invertible matrix $X$, such that $XA=AM$?</p>
| Community | -1 | <p>Some basic facts that can be used to reason about the problem:</p>
<ul>
<li>If $X$ is invertible, the rowspace of $XA$ is the rowspace of $A$.</li>
<li>Every row of $AM$ is in the row space of $M$.</li>
</ul>
|
330,488 | <p>On the complex plane <span class="math-container">$\mathbb C$</span> consider the half-open square <span class="math-container">$$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$</span> </p>
<p>Observe that for every <span class="math-container">$z\in \mathbb C$</span> and <span class="math-container"... | Taras Banakh | 61,536 | <p>Both problems have negative answer under CH and positive answer under <span class="math-container">$\neg$</span>CH. The proofs can be found <a href="https://arxiv.org/abs/1905.02243" rel="nofollow noreferrer">here</a>.</p>
|
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | Community | -1 | <p>Let me answer for any variable exponent $n(x)$: no,</p>
<p>$$(x^{n(x)})'=n(x) x^{n(x)-1}$$ doesn't hold.</p>
<p>As a counterexample, consider the function</p>
<p>$$n(x):=\frac1{\ln(x)}$$</p>
<p>remarking that $$x^{n(x)}=x^{1/\ln(x)}=e^{\ln(x)/\ln(x)}=e.$$</p>
<p>The wrong rule would yield</p>
<p>$$\frac1{\ln(x... |
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | Narasimham | 95,860 | <p>No, you cannot temporarily assume one of two variables to be constant when differentiating with respect exponent or argument in base function separately.</p>
<p>To handle both at the same time, take logarithm on both sides</p>
<p>$$\ln y =\sin x \ln x$$ and by the chain rule we get</p>
<p>$$\frac{y'}{y}=\cos x \l... |
2,372,035 | <p>$ABCD$ is a square with side-length $1$ and equilateral triangles $\Delta AYB$ and $\Delta CXD$ are inside the square. What is the length of $XY$?</p>
| Michael Rozenberg | 190,319 | <p>Just $2\cdot\frac{\sqrt3}{2}-1$</p>
|
629,887 | <p>I was going through one of the topic "Introduction to Formal proof". In one example while explaining "Hypothesis" and "conclusion" got confused.</p>
<p>The example is as follows:</p>
<blockquote>
<p>If $x\geq4$, then $2^x \geq x^2$.</p>
</blockquote>
<p>While deriving conclusion article said "As $x$ grows large... | Sasha | 11,069 | <p>Using
$$
\binom{n}{k} \frac{k}{n} = \binom{n-1}{k-1}
$$
Then
$$
\sum_{k=1}^n \frac{k}{n} \binom{n}{k} t^k (1-t)^{n-1} = \sum_{k=1}^{n} \binom{n-1}{k-1} t^{k} (1-t)^{n-k} = t \sum_{k=0}^{n-1} \binom{n-1}{k} t^{k} (1-t)^{n-1-k} = (t+1-t)^{n-1} t = t
$$</p>
|
4,219,492 | <p>This is a long question/explanation, where I more or less have a method to the madness; however, I've stumbled upon how I perform the calculation and I feel it is quite crude. This question is based off <a href="https://math.stackexchange.com/questions/2631501/finding-the-distribution-of-the-sum-of-three-independent... | Novice C | 81,628 | <p>Since this was an ill posed question, and I tried to throw in a concrete question at the end, I've answered it myself (as to not waste anyone's time further). The 8 integrals I got are:</p>
<p><span class="math-container">$$\begin{align}
1.& \int_0^z t(z-t)\ \textrm{d}t, &0 \leq z < 1,\\
2.& \int_{z-1... |
321,255 | <p>I've been looking for the definition of projective ideal but haven't found anything, all I've seen is the definition of projective module (but I don't know how these are related, if they are ¿?). Does anyone know a book where I can look up basic facts about projective ideals?</p>
| Boris Novikov | 62,565 | <p>A (left) projective ideal is just a projective submodule of the regular module ${}_RR$ (i.e. the ring $R$ considered as a left $R$-module). So if ${}_RR$ can be decomposed in a direct sum, then every summand is a projective ideal.</p>
|
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