qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | Magma | 599,209 | <p>Unlike the other answers to this question, I claim that it's possible to trick an explorer on an infinite plane into thinking he's on a sphere. In fact, I'm about to trick you, by providing just a screenshot of <a href="https://github.com/Ralith/hypermine" rel="noreferrer">a work-in-progress video game project</a> I... |
3,223,618 | <p>I have this system of linear equations with parameter:</p>
<p><span class="math-container">$ ax + 4y + z =0 $</span></p>
<p><span class="math-container">$2y + 3z = 1$</span> </p>
<p><span class="math-container">$3x -cz=-2$</span></p>
<p>What I did was to put those equations into a matrix and transform that matr... | Dr. Sonnhard Graubner | 175,066 | <p>From the first equation we get
<span class="math-container">$$z=-ax-4y$$</span> so we get with the third equation
<span class="math-container">$$-3ax-10y=1$$</span> and with the last one:
<span class="math-container">$$3x+acx+4yc=-2$$</span></p>
<p>From the second equation above <span class="math-container">$$y=-\f... |
636,246 | <blockquote>
<p>Let $g(x)=x^2\sin(1/x)$, if $x \neq 0$ and $g(0)=0$. If $\{r_i\}$ is the numeration of all rational numbers in $[0,1]$, define
$$
f(x)=\sum_{n=1}^\infty \frac{g(x-r_n)}{n^2}
$$
Show that $f:[0,1] \rightarrow R$ is differentiable in each point over [0,1] but $f'(x)$ is discontinuous over each $r_n$... | Cameron Williams | 22,551 | <p>First a stylistic comment: you should use the word "differentiable" in place of "derivable." Second: you should show that $f$ is well-defined on $[0,1]$ so that you can take its derivative (this is easy). We want to consider the difference quotient $\frac{f(x)-f(y)}{x-y}$ and what happens as $x\rightarrow y$ (I will... |
1,545,583 | <p>Suppose that $K$ is an infinite compact metric space. Define $c_0=\{ (x_n)_{n \in \mathbb{N}}| \lim_n{\| x_n \|}=0 \}$.</p>
<p>Is it true that $c_0$ complemented in $C(K)$, the set of continuous functions on $K$?</p>
<p>It seems true based on this <a href="http://www.ams.org/journals/proc/2006-134-04/S0002-9939-05... | Jochen | 38,982 | <p>Let $t_n \in K$ be a convergent sequence of distinct points with limit $t_\infty$ also distinct from all $t_n$, consider
$C_0(K)=\lbrace f\in C(K): f(t_\infty)=0\rbrace$ and consider $P:C_0(K)\to c_0$, $f\mapsto (f(t_n))_{n\in\mathbb N}$. Conversely, choose peak functions $\varphi_n \in C(K)$ with disjoint supports ... |
1,545,583 | <p>Suppose that $K$ is an infinite compact metric space. Define $c_0=\{ (x_n)_{n \in \mathbb{N}}| \lim_n{\| x_n \|}=0 \}$.</p>
<p>Is it true that $c_0$ complemented in $C(K)$, the set of continuous functions on $K$?</p>
<p>It seems true based on this <a href="http://www.ams.org/journals/proc/2006-134-04/S0002-9939-05... | Tomasz Kania | 17,929 | <p>Even more is true. Every copy of $c_0$ in $C(K)$ for $K$ compact, metric is complemented by a projection of norm at most 2. Indeed, $C(K)$ is in this case separable (as $K$ is second-countable we may use the <a href="https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" rel="nofollow">Stone–Weierstrass th... |
10,942 | <p>I heard about it sometime somewhere and want to read about it now, but I can't recall what the name is:</p>
<p>Start with $a_1 = \ldots =a_n=1$. Choose a number between 1 and $n$ with probability $a_i/(a_1+ \ldots + a_n)$ to choose $i$. If $i_0$ is the number chosen, increase $a_i$ by 1 and now choose another numbe... | Shai Covo | 2,810 | <p>For this process and a substantial generalization, see <a href="http://www.combinatorics.net/aoc/toc/v7n2/Chung/7_2_141.pdf" rel="nofollow">this</a>.
The case $p=0$ and $\gamma = 1$ corresponds to the process you described.</p>
|
302,061 | <p>Can you say how to find number of non-abelian groups of order n?</p>
<p>Suppose n is 24 ,then from structure theorem of finite abelian group we know that there are 3 abelian groups.But what can you say about the number of non-abelian groups of order 24?</p>
<p>The following link is a list of number of groups of or... | Andreas Caranti | 58,401 | <p>The point is, the numbers grow very fast, particularly for prime powers. Look at <a href="http://groupprops.subwiki.org/wiki/Groups_of_order_2%5En" rel="nofollow">this list</a> for groups of order $2^{k}$, for $k \le 10$. (I will try and provide a better reference later.)</p>
|
3,242,363 | <blockquote>
<p>Why does this function, <span class="math-container">$$\tan\left(x ^ {1/x}\right)$$</span>
have a maximum value at <span class="math-container">$x=e$</span>?</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/pqE0Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pqE0Q.png" ... | Vineet | 196,541 | <p>This is going to be clumsy.</p>
<p><span class="math-container">$a=(x+b)\cdot0.029+0.3 ~~~~~~~~~(i)\\
b=(x+a)\cdot0.015 ~~~~~~~~~~~~~~~~~~(ii)\\
\frac{b}{0.015}=x+a\Rightarrow \boxed{a = \frac{b}{0.015}-x}$</span></p>
<p>Putting this in equation <span class="math-container">$(i)$</span>,</p>
<p><span class="math-... |
2,012,318 | <p>Find the volume of:</p>
<p>$V=[(x,y,z): 0 \leqslant z \leqslant 4 - \sqrt{x^2+y^2}, 2x \leqslant x^2+ y^2 \leqslant 4x] $</p>
<p>I should somehow construct triple integral here in order to solve this, which means that i have to find limits of integration for three variables, but i am just not quite sure how, i ass... | Eugene Zhang | 215,082 | <p>\begin{align}
(x,y)\in ((A \times B) -(A\times C))&\iff (x,y)\in (A \times B) \land (x,y) \not \in (A \times C)
\\
&\iff(x\in A \land y\in B) \land (x\not \in A \lor y \not \in C)
\\
&\iff (x \in A \land y \in B \land x\not \in A) \lor (x \in A \land y \in B \land y\not \in C)
\\
&\iff 0\lor (x \in A... |
2,994,970 | <p>As far that i have known, i understand the notion "a function on the circle" by each one of the followings (both equivalent):</p>
<ol>
<li>A function is defined on <span class="math-container">$\mathbb{R}$</span> that is <span class="math-container">$2\pi-$</span>periodic.</li>
<li>A function that is defined on <sp... | edm | 356,114 | <p>The whole theory of Fourier series is built to study periodic functions. The correct interpretation of "integrable function on the circle" is 2.</p>
|
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | Daniel Huber | 46,318 | <p>You can make a hexagonal grid using only MMA built in functions. You may adapt the code to your liking:</p>
<pre><code>c3 = Cos[30 Degree]; s3 = Sin[30 Degree];
del1 = {Sqrt[c3^2 + Sqrt[(1 + s3^2)^2 + c3^3]], c3} // N;
del2 = {-Sqrt[c3^2 + Sqrt[(1 + s3^2)^2 + c3^3]], c3} // N;
del3 = {0, 2 c3};
trans[del_] := Map[(d... |
231,479 | <p>Is there a function that can create hexagonal grid?</p>
<p>We have square grid graph, where we can specify <code>m*n</code> dimensions:</p>
<pre><code>GridGraph[{m, n}]
</code></pre>
<p>We have triangular grid graph (which works only for argument <code>n</code> up to 10 - for unknown reason):</p>
<pre><code>GraphDat... | kglr | 125 | <p>We can generate the vertex coordinates using a slightly modified version of azerbajdan's <code>cells</code> and use them with <a href="https://reference.wolfram.com/language/ref/NearestNeighborGraph.html" rel="noreferrer"><code>NearestNeighborGraph</code></a>:</p>
<pre><code>ClearAll[vCoords]
vCoords = DeleteDuplica... |
1,028,720 | <p>I was wondering the following:</p>
<blockquote>
<p><strong>Background Question:</strong> Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented?</p>
</blockquote>
<p>By "$X_0$ is a copy of $X$", I mean $X_0 \cong X$ via an invertible, bounded, linear map. So... | Mike F | 6,608 | <p>Tomek's answer is very comprehensive. However, I thought I would add a "lower level" explanation.</p>
<blockquote>
<p><strong>Proposition:</strong> $\{0\} \oplus c_0 \oplus \ell_\infty \oplus \ell_\infty \oplus \ell_\infty \oplus \ldots$ is not complemented in $c_0 \oplus \ell_\infty \oplus \ell_\infty \oplus \el... |
4,236,077 | <p>Suppose that <span class="math-container">$A\subset B$</span> and <span class="math-container">$A\subset C$</span>. Why does this imply <span class="math-container">$A\subset B\cup C?$</span></p>
<p>If <span class="math-container">$x\in A$</span>, then since <span class="math-container">$A\subset B$</span> and <span... | Shaun | 104,041 | <p>This is trivial if <span class="math-container">$A=\varnothing$</span>.</p>
<p>Let <span class="math-container">$x\in A$</span>. If <span class="math-container">$A\subset B$</span>, then <span class="math-container">$x\in A$</span> implies <span class="math-container">$x\in B$</span>. If <span class="math-container"... |
4,236,077 | <p>Suppose that <span class="math-container">$A\subset B$</span> and <span class="math-container">$A\subset C$</span>. Why does this imply <span class="math-container">$A\subset B\cup C?$</span></p>
<p>If <span class="math-container">$x\in A$</span>, then since <span class="math-container">$A\subset B$</span> and <span... | ryang | 21,813 | <p>Continuing from where you left off:
<span class="math-container">\begin{aligned}x&\in B\cap C\\&\subset B\cup C.\end{aligned}</span></p>
<p>Therefore, since <span class="math-container">$x\in A\implies x\in B\cup C,$</span> <span class="math-container">$$A\subset B\cup C.$$</span></p>
|
210,110 | <p>A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n << 1$. Does this approximation have a name? Any leads on estimating the error of the approximation?</p>
| Christian Fries | 75,356 | <p>Wikipedia calls it <a href="https://en.wikipedia.org/wiki/Binomial_approximation" rel="nofollow noreferrer">Binomial Approximation</a>.</p>
|
117,285 | <p>Let $R \subseteq A \times A$ and $S \subseteq A \times A$ be two arbitary equivalence relations.
Prove or disprove that $R \cup S$ is an equivalence relation.</p>
<p>Reflexivity: Let $(x,x) \in R$ or $(x,x) \cup S \rightarrow (x,x) \in R \cup S$</p>
<p>Now I still have to prove or disprove that $R \cup S$ is symme... | dtldarek | 26,306 | <p>Transitivity fails. Let $A = \{1,2,3\}$, $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$ and $S = \{(1,1), (2,2), (3,3), (2,3), (3,2)\}$, then $R \cup S$ contains both $(1,2)$ and $(2,3)$, but not $(1,3)$.</p>
<p>The symmetry could be worded better, but is alright. The important thing is that if $(x,y) \in R \cup S$ th... |
4,074,630 | <p>Let <span class="math-container">$f: [a,b] \to [0,\infty)$</span> and <span class="math-container">$f$</span> is Riemann Integrable on every subinterval <span class="math-container">$[a + \epsilon,b]$</span> for <span class="math-container">$\epsilon > 0$</span>. Suppose that the improper Riemann integral exists.... | RRL | 148,510 | <p>Since <span class="math-container">$f$</span> is nonnegative, the Lebesgue integral must exist (but may be infinite). With the existence of the improper Riemann integral we can show that the Lebesgue integral is finite and <span class="math-container">$f$</span> is "Lebesgue integrable" on <span class="ma... |
293,937 | <p>Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$.</p>
<p>Now let me replace the absolutely continuous measures $f(x)dx$ and $g(x)dx$, by a pair $\lambda,\mu$ of non-negative bounded measures on $\Omega$... | Fedor Petrov | 4,312 | <p>Any two measures are a.c. with respect to their sum, and you may take the geometric mean of the densities. This is the same definition as yours with infimum. </p>
|
293,937 | <p>Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$.</p>
<p>Now let me replace the absolutely continuous measures $f(x)dx$ and $g(x)dx$, by a pair $\lambda,\mu$ of non-negative bounded measures on $\Omega$... | R W | 8,588 | <p>This is what is called "Hellinger integral" and appears in the definition of the <a href="https://en.wikipedia.org/wiki/Hellinger_distance" rel="nofollow noreferrer">Hellinger distance</a>.</p>
|
1,913,835 | <p>I'm having a difficult time explaining/understanding a (seemingly) simple argument of an algorithm that I know I can use to determine if a directed graph <strong>G</strong> is strongly connected.</p>
<p>The algorithm that I know (does this have a name?) goes like this:</p>
<pre><code>Use BFS (breadth-first-search)... | angryavian | 43,949 | <p>I think the analysis is simple if the starting note $S$ for the second run of BFS is the same as the starting node of the first run.</p>
<p>The first run tells you that every node can be reached from $S$; the second run tells you every node can reach $S$. Combining these two facts shows strong connectivity: you can... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Georges Elencwajg | 450 | <p>Many, many things have changed in the last 60 years. A mathematician of the fifties (in Europe) was required to know descriptive geometry, rational mechanics, maybe some astronomy, and a lot of physics. He (yes!) was supposed to know how to calculate rather difficult primitives and have many tricks at his fingertip... |
4,614,334 | <p>I'm trying to resolve this recurrence equation <span class="math-container">$T(n)=4T(\frac{n}{2})+cn$</span>.</p>
<p>The solution I discovered online is <span class="math-container">$T(n)=\theta(n^2)$</span>.</p>
<p>The steps I follow was these: <br/>
a) Create the tree of recurrence as this:</p>
<pre><code> ... | sibillalazzerini | 1,137,888 | <p><span class="math-container">$T(n)=4T(n/2)+cn$</span></p>
<p><span class="math-container">$ =4[4T(n/4)+cn/2]+cn$</span></p>
<p><span class="math-container">$ =16T(n/4)+3cn$</span></p>
<p><span class="math-container">$ =16[4T(n/8)+cn/4]+3cn$</span></p>
<p><span class="math-container">$ =64T(n/8)+7cn$</spa... |
4,614,334 | <p>I'm trying to resolve this recurrence equation <span class="math-container">$T(n)=4T(\frac{n}{2})+cn$</span>.</p>
<p>The solution I discovered online is <span class="math-container">$T(n)=\theta(n^2)$</span>.</p>
<p>The steps I follow was these: <br/>
a) Create the tree of recurrence as this:</p>
<pre><code> ... | zwim | 399,263 | <p>Rewrite <span class="math-container">$T(2n)=4T(n)+2cn$</span> and set <span class="math-container">$U(n)=\dfrac{4T(n)}{n}+\alpha \dfrac{2cn}n$</span></p>
<p>(this is motivated by this -> <a href="https://math.stackexchange.com/a/3002925/399263">https://math.stackexchange.com/a/3002925/399263</a>)</p>
<p>After sub... |
2,322,678 | <p>We have $n$ different elements $(a_1,...,a_n)$ that are all the elements of $K$ and $\in$ finite field $K$.
I want to prove, that $\prod_{i=1}^{n} (X - a_i) + 1 \in K[X]$ doesn't have roots</p>
<p>I know, that if $a_i$ is a root of polynomial $p \in K[X]$ , then exists $f \in K[X]$ such that $p = (x - a_i)f$</p>... | SvanN | 446,362 | <p>We know that $\lim_{x \to \infty} \frac{\sin{x}}{x} = 0$. Because the sine oscillates between being positive and negative, we can divide the integral up into 'chunks', on each of which the function is either wholly positive or wholly negative. If the integral exists, then this will be equal to the series of these 'c... |
2,322,678 | <p>We have $n$ different elements $(a_1,...,a_n)$ that are all the elements of $K$ and $\in$ finite field $K$.
I want to prove, that $\prod_{i=1}^{n} (X - a_i) + 1 \in K[X]$ doesn't have roots</p>
<p>I know, that if $a_i$ is a root of polynomial $p \in K[X]$ , then exists $f \in K[X]$ such that $p = (x - a_i)f$</p>... | Angina Seng | 436,618 | <p>Integration by parts gives
$$\int_0^N\frac{\sin x}{x}\,dx
=\left[\frac{1-\cos x}{x}\right]_0^N+\int_0^N\frac{1-\cos x}{x^2}\,dx
=\frac{1-\cos N}{N}+\int_0^N\frac{1-\cos x}{x^2}\,dx.$$
So
$$\lim_{N\to\infty}\int_0^N\frac{\sin x}{x}\,dx
=\lim_{N\to\infty}\int_0^N\frac{1-\cos x}{x^2}\,dx.$$
This last is a convergent in... |
2,322,678 | <p>We have $n$ different elements $(a_1,...,a_n)$ that are all the elements of $K$ and $\in$ finite field $K$.
I want to prove, that $\prod_{i=1}^{n} (X - a_i) + 1 \in K[X]$ doesn't have roots</p>
<p>I know, that if $a_i$ is a root of polynomial $p \in K[X]$ , then exists $f \in K[X]$ such that $p = (x - a_i)f$</p>... | DonAntonio | 31,254 | <p>About your last question in your post:</p>
<p>$$\lim_{x\to\infty}-\frac{\cos c}c=-\lim_{c\to\infty}\,\frac1c\cdot\cos c=0$$</p>
<p>since the last is the limit of a function whose limit zero times a <strong>bounded</strong> one.</p>
|
2,595,418 | <p>Let there be a graph $G$ and it's complement $G'$ , if the degree of a vertex in $G$ is added with degree of the corresponding vertex of $G'$ , the sum will be $(n-1)$; where $n$ is the number of vertices. How to prove this ?</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>The union of $G$ and $\overline G$ contains an edge between each pair of the $n$ vertices. Hence, the union is $K_n$ which is the <a href="https://en.m.wikipedia.org/wiki/Complete_graph" rel="nofollow noreferrer">complete graph</a> of $n$ vertices. </p>
<p>What is the degree of each v... |
14,140 | <p>One of the most annoying "features" of <em>Mathematica</em> is that the <code>Plot</code> family does extrapolation on <code>InterpolatingFunction</code>s without any warning. I'm sure it was discussed to hell previously, but I cannot seem to find any reference. While I know how to simply overcome the problem by def... | Rojo | 109 | <p>As you can see with </p>
<pre><code>if = Interpolation[Range[10]];
With[{mess = $Messages},
Plot[
Block[{$Messages = mess}, if[x]], {x, 5, 20}, Evaluated -> False]
]~Quiet~Message::msgl
</code></pre>
<p>the problem is that <code>$Messages</code> gets blocked somehow. I have so far no good recomme... |
3,587,891 | <blockquote>
<p>Suppose <span class="math-container">$f(x, y, z): \mathbb{R}^{3} \rightarrow \mathbb{R}$</span> is a <span class="math-container">$C^{2}$</span> harmonic function, that is, it satisfies <span class="math-container">$f_{x x}+f_{y y}+f_{z z}=0 .$</span> Let <span class="math-container">$E \subset \mathbb{... | Henry Swanson | 55,540 | <p>Yeah, that's the approach you're supposed to take, I think.</p>
<p>Picking up where you left off, you want to show that, given two sections <span class="math-container">$f_i \in \mathscr F(U_i)$</span>, <span class="math-container">$f_j \in \mathscr F(U_j)$</span>, that they agree when restricted to <span class="mat... |
3,046,083 | <p>Is it true that the intersection of the closures of sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> is equal to the closure of their intersection?
<span class="math-container">$ cl(A)\cap{cl(B)}=cl(A\cap{B})$</span> ?</p>
| bubba | 31,744 | <p>No, it's not true. Look at <a href="https://en.wikipedia.org/wiki/Closure_(topology)#Facts_about_closures" rel="nofollow noreferrer">this page</a>, or <a href="https://math.stackexchange.com/questions/1851554/closure-of-intersection-is-subset-of-intersection-of-closures?rq=1">this question</a>.</p>
<p>A simple coun... |
3,779,785 | <p>So I have this problem, <span class="math-container">$W=3^n -n -1$</span>. How to find all <span class="math-container">$n$</span> so <span class="math-container">$W$</span> can be divided by <span class="math-container">$5$</span>.</p>
<p><em>what I tried:</em>
I found all the remainders of <span class="math-contai... | lhf | 589 | <p>Let <span class="math-container">$W_n=3^n -n -1$</span>.
Then <span class="math-container">$W_n \bmod 5$</span> is periodic of period <span class="math-container">$20$</span>:
<span class="math-container">$$
0,1,1,3,1,2,2,4,2,3,3,0,3,4,4,1,4,0,0,2,0,1,1,3,\dots
$$</span></p>
<p>This follows by induction because <spa... |
3,773,856 | <p>I'm having trouble with part of a question on Cardano's method for solving cubic polynomial equations. This is a multi-part question, and I have been able to answer most of it. But I am having trouble with the last part. I think I'll just post here the part of the question that I'm having trouble with.</p>
<p>We ha... | Community | -1 | <p><strong>An alternative method:</strong></p>
<p>We can try to turn
<span class="math-container">$$t^3+pt+q=0$$</span>
into
<span class="math-container">$$4\cos^3\theta-3\cos\theta=a$$</span> by a change of variable: we set</p>
<p><span class="math-container">$$t=\lambda \cos\theta$$</span> and solve</p>
<p><span clas... |
3,795,234 | <p>(disclaimer: I am not well versed in mathematics so please excuse my poor notation / explanation)</p>
<p>Given a hexagon grid that defines it's "neighbours" via offsets on the axis' <span class="math-container">$q$</span> & <span class="math-container">$r$</span> like this :
<a href="https://i.stack.im... | Somos | 438,089 | <p>Given hexagon integer coordinates <span class="math-container">$\,q\,$</span> and <span class="math-container">$\,r,\,$</span> define <span class="math-container">$\,s:=-q-r\,$</span>
which implies <span class="math-container">$\,0=q+r+s.\,$</span> Two hexagons are "neighbors" exactly when one
of the <span... |
3,012,090 | <p>Let <span class="math-container">$x>0$</span>. I have to prove that</p>
<p><span class="math-container">$$
\int_{0}^{\infty}\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos(p\frac{\pi}{2})}\tag{1}
$$</span></p>
<p>by converting the integral on the left side to a double integral using the expression below:</p>
<p... | Yadati Kiran | 490,720 | <p>Hint: <span class="math-container">$\displaystyle\int_{0}^{\infty}\frac{\cos x}{x^p}dx= \text{Real part of}\:\int_{0}^{\infty}\frac{e^{iz}}{z^p}dz$</span> and use residue theorem. This has a pole of order <span class="math-container">$p$</span> hence the term <span class="math-container">$\Gamma (p)$</span> in the d... |
3,012,090 | <p>Let <span class="math-container">$x>0$</span>. I have to prove that</p>
<p><span class="math-container">$$
\int_{0}^{\infty}\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos(p\frac{\pi}{2})}\tag{1}
$$</span></p>
<p>by converting the integral on the left side to a double integral using the expression below:</p>
<p... | omegadot | 128,913 | <p>So let us follow your initial line of thought and convert the integral to a double integral. As you correctly observe, as
<span class="math-container">$$\frac{1}{x^p} = \frac{1}{\Gamma (p)} \int_0^\infty e^{-xt} t^{p - 1} \, dt,$$</span>
which, by the way, is just the Laplace transform for the function <span class="... |
2,837,934 | <p>A cyclist gets left behind by $500$ meters every $minute$ by motorcyclist, because of that he takes $2$ $hour$ and $42$ $minute$ more than motorcyclist to cover $52$ $km$.
Find both of their speed.</p>
<p>My approach: $v_2-v_1=30km/h$ (converted 500 meter per minute to km/h)</p>
<p>$v_2=52/t$<br>
$v_1=52/(t+2.42)$... | fleablood | 280,126 | <p>$500 \frac {meters}{minute} * \frac {km}{1000 meters}*\frac {60 minutes}{hr}= 30 \frac {km}{hr}$</p>
<p>$2 hr 42 minutes = 2\frac {42}{60} hr = 2\frac {7}{10} = 2.7 hr$.</p>
<p>$v_2 - v_1 = 30\frac {km}{hr}$ </p>
<p>$v_2 = \frac {52}{t}$</p>
<p>$v_1 = \frac {52}{t + 2.7}$ so </p>
<p>$\frac {52}{t}-\frac {52}{t ... |
2,375,529 | <p>Let $H$ be a Hilbert space and let $T\in \mathcal{B}(H)$ such that $T$ is self-adjoint. I want to show that if $T$ is non-zero, then $T^n\neq 0$ for all $n\in \mathbb{N}$.</p>
<p>Suppose $n$ be the least positive integer such that $T^n=0$. Then for all $x,y\in H$, we have $\langle T^nx,y\rangle=0\implies \langle T^... | Angina Seng | 436,618 | <p>If $n$ is even then $0=\left<T^n x,x\right>=\left<T^{n/2} x,T^{n/2}x\right>$
so $T^{n/2}x=0$. What if $n$ is odd?</p>
|
1,671,111 | <p>I'm looking for an elegant way to show that, among <em>non-negative</em> numbers,
$$
\max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\}
$$</p>
<p>I can show that $\max \{a+b, c+d\} \leq \max \{a,c\} + \max \{b,d\}$ by exhaustively checking all possibilities of orderings am... | marty cohen | 13,079 | <p>More than this is true.</p>
<p>Let $P$ be a permutation of
$[1, 2, ..., n]$.</p>
<p>Then
$\max \{a_1 + b_{P(1)}, \dots, a_n + b_{P(n)}\}
\leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\}
$.</p>
<p>This is proved
in the same way
as carmichael561's proof:</p>
<p>For all $i$ from
$1$ to $n$,
$a_i \le \max \... |
3,075,263 | <p>Let <span class="math-container">$A$</span> be a positive semi-definite matrix. How to show that Frobenius norm is less than trace of the matrix? Formally,
<span class="math-container">$$\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$$</span>
Also, show when <span class="math-container">$A$</span> is an <span class="math-c... | angryavian | 43,949 | <p>Let <span class="math-container">$\sigma_1, \ldots, \sigma_r$</span> be the singular values of <span class="math-container">$A$</span>.
Then
<span class="math-container">$$\sqrt{\text{Tr}(A^\top A)} = \sqrt{\sum_i \sigma^2_r} \le \sum_i |\sigma_r| = \|A\|_*.$$</span></p>
|
2,629,408 | <p>How to evaluate this given expression?
$$\int\frac{du}{\sqrt{9e^{-2u}-1}}$$
I got so many tries but I'm not sure of my answer because somebody said that it was wrong, they told me that I used a wrong formula applied!
That's why I ask a support here I want correct explanation and answer of this given!</p>
<p>Thanks!... | Claude Leibovici | 82,404 | <p>Considering $$I=\int\frac{du}{\sqrt{9e^{-2u}-1}}$$ use
$$\sqrt{9e^{-2u}-1}=t\implies u=-\frac{1}{2} \log \left(\frac{1}{9} \left(t^2+1\right)\right)\implies du=-\frac{t}{t^2+1}\,dt$$ This makes
$$I=-\int\frac{dt}{t^2+1}$$</p>
|
3,454,095 | <p>Minimize <span class="math-container">$\;\;\displaystyle \frac{(x^2+1)(y^2+1)(z^2+1)}{ (x+y+z)^2}$</span>, if <span class="math-container">$x,y,z>0$</span>.
By setting gradient to zero I found <span class="math-container">$x=y=z=\frac{1}{\displaystyle\sqrt{2}}$</span>, which could minimize the function.</p>
<bl... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$x=\frac{a}{\sqrt2},$</span> <span class="math-container">$y=\frac{b}{\sqrt2}$</span> and <span class="math-container">$z=\frac{c}{\sqrt2}.$</span></p>
<p>Thus, since we can assume that <span class="math-container">$(a^2-1)(b^2-1)\geq0,$</span> by C-S we obtain:
<span class="math-co... |
2,078,535 | <p>I'm kinda new with this and find hard to solve the problems related to LA although I can visually and conceptualize stuff easily. Please help me to find $\{(v_1, v_2, v_3) \in \Bbb R^3 \mid 5v_1 - 3v_2 + 2v_3 = 0\}$.</p>
| hamam_Abdallah | 369,188 | <p>A vector $u=(v_1,v_2,v_3)$ which satisfies the condition $: 5v_1-3v_2+2v_3=0$ can be written as</p>
<p>$$u=(v_1,v_2,\frac{1}{2}(-5v_1+3v_2))$$
$$=v_1(1,0,-\frac{5}{2})+v_2(0,1,\frac{3}{2})$$</p>
<p>Put $ u_1=(1,0,-\frac{5}{2})$ and
$u_2=(0,1,\frac{3}{2})$. </p>
<p>$u_1$ and $u_2$ are independent so your space is... |
151,956 | <p>I'm looking for a general method to evaluate expressions of the form</p>
<p>$$\frac{\mathrm{d}(u^v)}{\mathrm{d}u}\text{ and }\frac{\mathrm{d}(u^v)}{\mathrm{d}v}\;.$$</p>
<p>I know that the answers to these are, respectively, $u^{v-1}v$ and $u^v\mathrm{ln}u$, but am unsure of how to obtain them, and how the chain r... | Brian M. Scott | 12,042 | <p>If $v$ is a constant, $\frac{d}{du}u^v=vu^{v-1}$, but the chain rule is not required. If $v$ is a function of $u$, then your formula is simply wrong: $u^v=\left(e^{\ln u}\right)^v=e^{v\ln u}$, so</p>
<p>$$\frac{d}{du}u^v=\frac{d}{du}e^{v\ln u}=e^{v\ln u}\left(\frac{v}u+\ln u\frac{dv}{du}\right)=u^v\left(\frac{v}u+\... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Buschi Sergio | 6,262 | <p>Of course your "intuition" request can be only about Set-based category where limits and colimits are based on Set analogue.</p>
<p>ABout Colimit you can think as a "amalgamated" union like glueing for a descent data (see also Boubaky- Topology (I vol.)).</p>
<p>ABout limits, is different, limits belong to the pro... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Anton Fetisov | 10,605 | <p>There's an enlightening example of limit coming from topology. Arguably it was one of the motivating examples for the notion of categorical limit. In general topology it is known as limit over a filter of subsets.</p>
<p>Consider a category $Ouv_X$ of open subsets for a topological space $X$, morphisms being the ob... |
96,369 | <p>Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?)</p>
<p>The only thing I was able to find was that some authors use the name extreme limits; see google books: <a href="http://www.google.com/search?tbm=bks&tbo=1&... | leslie townes | 18,076 | <p>For what it is worth I do not think there is any well-established terminology for what you want. I have never heard of any, anyway.</p>
<p>I think it is worthwhile to change your question to a slightly broader question: "what kinds of common generalizations could one make of these ideas, so that they become di... |
96,369 | <p>Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?)</p>
<p>The only thing I was able to find was that some authors use the name extreme limits; see google books: <a href="http://www.google.com/search?tbm=bks&tbo=1&... | Martin Sleziak | 8,297 | <p>It seems that the term <em>extreme limits</em> is used, although not very frequently.</p>
<p>Some people call limit superior and limit inferior <em>inner and <a href="https://en.wikipedia.org/wiki/Outer_limit" rel="nofollow noreferrer">outer limits</a></em>, see e.g. this spikedmath comic <a href="http://spikedmath... |
181,000 | <p>Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices.
Which circulant graphs over prime number of vertices greater than or equal to 11 are determined by the spectrum?</p>
| Brendan McKay | 9,025 | <p>No, and you don't need $p\ge 11$ either.</p>
<p>B Elspas, J Turner
Graphs with circulant adjacency matrices,
J. Combinatorial Theory, 9 (1970), pp. 297–307.</p>
<p>This paper shows that no two non-isomorphic circulant graphs on a prime number of vertices have the same spectrum. It doesn't say anything about circu... |
181,000 | <p>Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices.
Which circulant graphs over prime number of vertices greater than or equal to 11 are determined by the spectrum?</p>
| Shahrooz | 19,885 | <p>We know that all groups with prime order is CI-group. So, if two circulant graph with $p$ vertices be cospectral, then they are isomorphic. So, as Dear Brendan said, the answer is no.</p>
<p>But about your comment, the answer is yes. Just look at strongly regular graphs of order $29$. In this order we have Paley gr... |
295,597 | <p>I'm trying to solve this simple integral:</p>
<p>$$\frac12 \int \frac{x^2}{\sqrt{x + 1}} dx$$</p>
<p>Here's what I have done so far:</p>
<ol>
<li><p>$\displaystyle t = \sqrt{x + 1} \Leftrightarrow x = t^2 - 1 \Rightarrow dx = 2t dt$</p></li>
<li><p>$\displaystyle \frac12 \int \frac{x^2}{\sqrt{x + 1}} dx = \int \f... | Tapu | 17,142 | <p>No, everything is fine. Now simplify what you have got after integration. Note that $\sqrt{a^5}=\sqrt{a}.a^2$ etc.</p>
|
2,900,372 | <p>Let $R=\mathbb{Z}[\sqrt{-2}]$</p>
<blockquote>
<p>1) Is it true that for every free $R-$module $M$ of rank=$n$ that $M$ is a free $\mathbb{Z}-$module of rank=$2n$?</p>
<p>2) Find two non-isomorphic $R$-modules with $19$ elements each.</p>
</blockquote>
<p>For (1) I tried to begin with a basis $\{m_1,...,m_n... | Sarvesh Ravichandran Iyer | 316,409 | <p>Let $M = [m_1,...,m_n]$. Then, every element of $M$ can be written uniquely as $\sum a_im_i$ where $a_i \in \mathbb Z[\sqrt -2] = b_i + c_i \sqrt{-2}$.</p>
<p>Clearly, $M$ is then generated by $[m_i, m_i\sqrt{-2}]$ over $\mathbb Z$.The question is : is this set linearly independent over $\mathbb Z$?</p>
<p>Well, ... |
56,134 | <p>As mentioned,
I wish to read the first line of a file, and if needed, overwrite it with a new string.
The aim is to have a CSV with a list of possible elements.
Example:</p>
<pre><code>Adding the elements: 5 A, 6 B, and 7 C to a blank CSV:
A B C
5 6 7
Adding 4 A, 9 D:
A B C D
5 6 7
4 0 0 9
Adding 2 B, 7 E
A B C D... | Nick Lariviere | 18,824 | <p>Low-level file operators like Write won't work here because OutputStreams (such as you get with OpenWrite and OpenAppend) can't have their StreamPosition set before the end of the file. In general overwriting characters in an existing file isn't terribly trivial; you can use c-functions like<code>file_ptr = fopen(f... |
419,625 | <p>Please help me, I have two functions:</p>
<blockquote>
<pre><code>a := x^3+x^2-x;
b := 20*sin(x^2)-5;
</code></pre>
</blockquote>
<p>and I would like to change a background color and fill the areas between two curves. I filled areas but I dont know how can I change the background, any idea?</p>
<blockquote>
<pre>... | acer | 12,448 | <p>One way is to add a colored rectangle to the displayed items. Note that the filled red region now gets transparency=0.0 so that the green doesn't bleed through it.</p>
<pre><code>a := x^3+x^2-x:
b := 20*sin(x^2)-5:
plots[display](
plottools[transform]((x, y)-> [x, y+x^3+x^2+x])
(plot(20*sin(x^2)-5-x... |
2,389,782 | <p>Other than <a href="http://rads.stackoverflow.com/amzn/click/3885380064" rel="nofollow noreferrer">Engelking General Topology</a>, I also come across other graduate general topology text such as <a href="http://rads.stackoverflow.com/amzn/click/0697068897" rel="nofollow noreferrer">Dugundji</a> and <a href="http://w... | Alex Ravsky | 71,850 | <p>As far as I know, no. There are big general topology books “Topology” by Kuratowski (available at LibGen) and “Handbook of set-theoretic topology” (eds. Kunen and Vaughan) (also available in the Internet), but they consider selected topics. </p>
|
2,389,782 | <p>Other than <a href="http://rads.stackoverflow.com/amzn/click/3885380064" rel="nofollow noreferrer">Engelking General Topology</a>, I also come across other graduate general topology text such as <a href="http://rads.stackoverflow.com/amzn/click/0697068897" rel="nofollow noreferrer">Dugundji</a> and <a href="http://w... | William Elliot | 426,203 | <p>I like <em>General Topology</em> by Willard for reference of difficult theorems.<br>
It is point set topology orientated instead of analysis orientated.</p>
|
3,965,834 | <p>Does this sum converge or diverge?</p>
<p><span class="math-container">$$ \sum_{n=0}^{\infty}\frac{\sin(n)\cdot(n^2+3)}{2^n} $$</span></p>
<p>To solve this I would use <span class="math-container">$$ \sin(z) = \sum \limits_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{(2n+1)!} $$</span></p>
<p>and make it to <span class="math... | CHAMSI | 758,100 | <p>Notice that for whatever <span class="math-container">$ p\geq 2 $</span>, <span class="math-container">$ \lim\limits_{n\to +\infty}{\left|\frac{n^{p}\sin{n}\left(n^{2}+3\right)}{2^{n}}\right|}=0 $</span>.</p>
<p>Thus <span class="math-container">$ \left|\frac{\sin{n}\left(n^{2}+3\right)}{2^{n}}\right|=\underset{\ove... |
4,201,477 | <blockquote>
<p>Integrate <span class="math-container">$$\int \frac{\cos 2x}{(\sin x+\cos x)^2}\mathrm dx$$</span></p>
</blockquote>
<p>I was integrating my own way.</p>
<p><span class="math-container">$$\int \frac{\cos 2x}{\sin^2x+2\sin x\cos x+cos^2}\mathrm dx$$</span>
<span class="math-container">$$\int \cot 2x \mat... | Henry Lee | 541,220 | <p>Here is another alternative:
<span class="math-container">$$\frac{\cos 2x}{(\cos x+\sin x)^2}=\frac{\cos^2 x-\sin^2x}{(\cos x+\sin x)^2}\\
=\frac{(\cos x+\sin x)(\cos x-\sin x)}{(\cos x+\sin x)^2}=\frac{\cos x-\sin x}{\cos x+\sin x}\\
=\frac{(\cos x+\sin x)'}{(\cos x+\sin x)}$$</span>
now sub <span class="math-conta... |
2,146,457 | <p>Simplify the ring $\mathbb Z[\sqrt{-13}]/(2)$. I have so far:</p>
<p>$$\mathbb Z[\sqrt{-13}]/(2) \cong \mathbb Z[x]/(2, x^2 + 13) \cong \mathbb Z_2[x] / (x^2 + 1)$$</p>
<p>Now how do I simplify it further? I know that $x^2 + 1 = (x + 1)^2$ in $\mathbb Z_2[x]$, is this useful?</p>
| user416819 | 416,819 | <p>Make the substitution $y = x + 1$ to get an isomorphism $(Z/2Z)[x] \cong (Z/2Z)[y]$. The answer is $(Z/2Z)[y]/(y^2)$.</p>
|
1,130,487 | <p>Jessica is playing a game where there are 4 blue markers and 6 red markers in a box. She is going to pick 3 markers without replacement.
If she picks all 3 red markers, she will win a total of 500 dollars. If the first marker she picks is red but not all 3 markers are red, she will win a total of 100 dollars. Under ... | Emanuele Paolini | 59,304 | <p>Nothing is wrong. You have found two different antiderivatives of the same function.
You know that antiderivatives differ only by a constant on each interval where they are defined... you can check that this is the case in your computation.</p>
|
1,393,869 | <p>Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has exactly three distinct real roots. I am wondering if there is a way to change variables to simplify this problem and... | Travis Willse | 155,629 | <p>Suppose that (including multiplicity) the roots of <span class="math-container">$$f(x) = A x^3 + B x^2 + C x + D,$$</span> <span class="math-container">$A \neq 0$</span>, are <span class="math-container">$r_1, r_2, r_3$</span>. Consider the quantity
<span class="math-container">$$\Delta(f) := A^4 (r_3 - r_2)^2 (r_1 ... |
759,810 | <p>I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?</p>
| Sandeep Silwal | 138,892 | <p>$\phi(25) = 20$ so $2012^{2012} \equiv 2012^{12} \pmod {25}$.</p>
<p>Then $2012^{12} \equiv 12^{12} \equiv 144^6 \equiv 36^3 \equiv 6 \pmod {25}$. </p>
<p>Then solving the system $N \equiv 6 \pmod {25}, \equiv 0 \pmod 4$ gives $n \equiv 56 \pmod {100}$. </p>
|
3,982,937 | <p>To avoid typos, please see my screen captures below, and the red underline. The question says <span class="math-container">$h \rightarrow 0$</span>, thus why <span class="math-container">$|h|$</span> in the solution? Mustn't that <span class="math-container">$|h|$</span> be <span class="math-container">$h$</span>?</... | Ben Grossmann | 81,360 | <p>Note that the definition of a limit involves the inequality <span class="math-container">$0 < |x - a| < \delta$</span>. Substituting <span class="math-container">$a = 0$</span> and <span class="math-container">$x = h$</span> yields
<span class="math-container">$$
0 < |h| < \delta,
$$</span>
which is the ... |
3,497,679 | <p>I was given this function:
<span class="math-container">$$
f(x)=
\begin{cases}
x+x^2, & x\in\Bbb Q\\
x, & x\notin \Bbb Q
\end{cases}
$$</span>
I first proved that it is continuous at <span class="math-container">$x=0$</span>.</p>
<p>Now I need to prove that that for every <span class="math-container">$x_0 \... | davidlowryduda | 9,754 | <blockquote>
<p>I know that I need to start by assuming that the limit does exist but I don't know how to reach a contradiction.</p>
</blockquote>
<p>That's not necessary. It is sufficient to simply prove that the limit doesn't exist.</p>
<p>And the easiest way to do that is to consider a limit along rational value... |
3,497,679 | <p>I was given this function:
<span class="math-container">$$
f(x)=
\begin{cases}
x+x^2, & x\in\Bbb Q\\
x, & x\notin \Bbb Q
\end{cases}
$$</span>
I first proved that it is continuous at <span class="math-container">$x=0$</span>.</p>
<p>Now I need to prove that that for every <span class="math-container">$x_0 \... | pre-kidney | 34,662 | <p>If <span class="math-container">$\lim_{x\to x_0}f(x)$</span> exists, then for any sequence of numbers <span class="math-container">$x_1,x_2,\ldots$</span> converging to <span class="math-container">$x_0$</span>, we must have that <span class="math-container">$f(x_1),f(x_2),\ldots$</span> converges to <span class="ma... |
3,497,679 | <p>I was given this function:
<span class="math-container">$$
f(x)=
\begin{cases}
x+x^2, & x\in\Bbb Q\\
x, & x\notin \Bbb Q
\end{cases}
$$</span>
I first proved that it is continuous at <span class="math-container">$x=0$</span>.</p>
<p>Now I need to prove that that for every <span class="math-container">$x_0 \... | Michael Hardy | 11,667 | <p>Suppose <span class="math-container">$x_0\ne0$</span> and there is a limit:
<span class="math-container">$$
L = \lim_{x\,\to\, x_0} f(x).
$$</span>
Let <span class="math-container">$\varepsilon = x_0^2/4.$</span> Then:</p>
<ul>
<li>There is some number <span class="math-container">$\delta_1>0$</span> such that f... |
2,202,339 | <p>It's easy to prove $x^2+1$ is never divisible by $4k+3$ primes. I know a non-constructive proof for existing $x$ so that $p|x^2+1$ for $4k+1$ primes. is there any constructive one?</p>
| aras | 277,932 | <p>First note that</p>
<p>$(p-1)! = -1 \mod p$</p>
<p>by Wilson's Theorem. Since $p = 4k+ 1$, we can write this product as</p>
<p>$1\cdot 2 \cdots (\frac{p-1}{2}) (\frac{p+1}{2}) \cdots (p-1) \equiv -1 \mod p$</p>
<p>Note that this can be written as</p>
<p>$1 \cdot 2 \cdots (\frac{p-1}{2}) \cdot (p-\frac{p-1}{2}) ... |
3,020,365 | <p>Let <span class="math-container">$A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$</span>.</p>
<p>Then </p>
<ol>
<li><p><span class="math-container">$\sup (A)<\frac{2}{\pi}+\frac{1}{n\pi}$</span> for all <span class="math-container">$n\ge 1$</span>.</p></li>
<li><p><span class="math-container">$\inf (A)>... | Mostafa Ayaz | 518,023 | <p>If the option <span class="math-container">$2$</span> is true, it means that <span class="math-container">$$\inf A\ge -{2\over 3\pi}$$</span> since <span class="math-container">$\inf A> -{2\over 3\pi}-{1\over n\pi}$</span> is true for all <span class="math-container">$n>1$</span>. Since <span class="math-conta... |
16,749 | <p>I wanted to remove the <code>Ticks</code> in my coding but i can't. Here when i try to remove the <code>Ticks</code> the number also gone. I need numbers without <code>Ticks</code>, <code>Ticks</code> and <code>GridLines</code> should be automatic and don't use<code>PlotRange</code> .</p>
<pre><code>BarChart[{{1,... | Mr.Wizard | 121 | <p>Simply use option:</p>
<pre><code>TicksStyle -> Directive[20, Opacity[0], FontOpacity -> 0.3]
</code></pre>
<p><img src="https://i.stack.imgur.com/yuedn.png" alt="Mathematica graphics"></p>
<p>Credit to <a href="https://mathematica.stackexchange.com/users/5/rm-rf">R.M</a> for showing me <code>FontOpacity</c... |
425,460 | <p>While browsing through several pages of nlab(mainly on n-Categories), I encountered the notion "foo" several times. However, there seems to be article on nlab about this notion. Is this some kind of category theorist slang? Please explain to me what this term means.</p>
| Haizum Skallah | 86,428 | <p>1.1 More than the sum of their parts.</p>
<p>We motivate this first chapter by noticing that while many real-world structures are compositional, the results of observing them are often not. The reason is that observation is inherently “lossy”: in order to extract information from something, one must drop the details... |
1,749,128 | <p>How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the procedure?</p>
| Doug Spoonwood | 11,300 | <p>According to Wikipedia, Alfred Tarski came up with a set of axioms for a substantial fragment of Euclidean Geometry. The axioms require <a href="https://en.wikipedia.org/wiki/Tarski's_axioms" rel="nofollow">no set theory</a>. The axioms are first-order, stand as precise, and have gotten explored in an automate... |
2,957,315 | <p><span class="math-container">$j_{1,1}$</span> denotes the first zero of the first Bessel function of the first kind. (That's a lot of firsts!) It's approximately equal to <span class="math-container">$3.83$</span>. My question is, is there any closed form expression for its value? Even a infinite series or infin... | Claude Leibovici | 82,404 | <p>If you want an <em>approximation</em> of the first root without invoking at any time the Bessel functions, what you could consider is an <span class="math-container">$[2m+1,2n]$</span> Padé approximation of <span class="math-container">$J_1(x)$</span> built at <span class="math-container">$x=0$</span>. It will be lo... |
259,734 | <p>Basically, what the title says. </p>
<p>Presumably, one could use the fact that monoidal categories (resp. strict monoidal categories) are one-object bicategories (resp. 2-categories) and use the Lack model structure on those, but I am unsure if this would work or not.</p>
| Karol Szumiło | 12,547 | <p>I'm not sure about the case of general monoidal categories. (Although I seem to recall a remark that there is no such structure since the category of monoidal categories is not cocomplete and a suitable replacement would be the category of multicategories. Perhaps somebody can confirm this.)</p>
<p>However, the cas... |
3,423,225 | <p>I know that the angle <span class="math-container">$\theta$</span> of a right-angled triangle, centered at the origin, is defined as the radian measure of its intersection point with the unit circle, and that <span class="math-container">$\cos(\theta)$</span> and <span class="math-container">$\sin(\theta)$</span> ar... | Dr. Richard Klitzing | 518,676 | <p>It simply is the <strong>theorem on intersecting lines</strong> (aka on confocal rays, or aka about similar triangles), which provides
<span class="math-container">$$\frac{A}{\cos(\theta)}=\frac{C}1=\frac{B}{\sin(\theta)}$$</span>
Solving the right or left equation then provides the searched for relations.</p>
<p>-... |
3,802,269 | <p>Evaluate <span class="math-container">$\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt$</span></p>
<p>My approach:</p>
<p><span class="math-container">$$
\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt = \lim_{x\rightarrow \infty} \frac{\int_{0}^{x}e^{t^2}dt}{x^{-1}e^{x^2}}
$$</span></p>
<p>Both the nume... | Eric Towers | 123,905 | <p><span class="math-container">\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} x^{-1} \mathrm{e}^{x^2}
&= \frac{\mathrm{d}}{\mathrm{d}x}(x^{-1})\mathrm{e}^{x^2} + x^{-1} \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^{x^2} \\
&= -x^{-2}\mathrm{e}^{x^2} + x^{-1}2x\mathrm{e}^{x^2} \text{,}
\end{align*}</span... |
3,802,269 | <p>Evaluate <span class="math-container">$\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt$</span></p>
<p>My approach:</p>
<p><span class="math-container">$$
\lim_{x\rightarrow \infty} x\int_{0}^{x}e^{t^2-x^2}dt = \lim_{x\rightarrow \infty} \frac{\int_{0}^{x}e^{t^2}dt}{x^{-1}e^{x^2}}
$$</span></p>
<p>Both the nume... | Mark Viola | 218,419 | <p>The numerator after applying L'Hospital's Rule in the OP is incorrect as it is written. Note that <span class="math-container">$$\frac{d}{dx} \int_0^x e^{t^2}\,dt=e^{x^2}\ne e^{x^2}-1$$</span></p>
<hr />
<p>Applying L'Hospital's Rule reveals that</p>
<p><span class="math-container">$$\begin{align}
\lim_{x\to \infty ... |
3,056,121 | <p>I'm trying to find a function with infinitely many local minimum points where x <span class="math-container">$\in$</span> [0,1] and f has only 1 root. No interval should exist where the function is constant.</p>
| Zacky | 515,527 | <p>For the first part of the question we can substitute <span class="math-container">$x=\frac{1}{t}$</span> in order to get: <span class="math-container">$$I=\int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x=\int^0_\infty \frac{\ln\left(\frac{1}{t}\right)}{t^2\left(1+\frac{1}{t^2}\right)^3}\frac{-dt}{t^2}=\int_0^\in... |
3,056,121 | <p>I'm trying to find a function with infinitely many local minimum points where x <span class="math-container">$\in$</span> [0,1] and f has only 1 root. No interval should exist where the function is constant.</p>
| Jia Ming جيا ميڠ | 383,520 | <p>I found a way to directly evaluate both of those integrals!</p>
<p>We first use the well known result from <a href="https://math.stackexchange.com/questions/290200/int-0-infty-frac-ln-xx2a2-mathrmdx-evaluate-integral"><span class="math-container">$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$</span> Evaluate Int... |
138,921 | <p>If $A\colon H\to S$ is a bounded operator on a Hilbert space $H$, and $S\subset H$. It is known that $\operatorname{trace}(A)=\sum_{n} \langle Af_n,f_n\rangle$ for any orthonormal basis $\{f_{n}\}$. Is there a relation between $\operatorname{trace}(A)$, $\operatorname{rank}(A)$, and dimension of $\operatorname{range... | Shuhao Cao | 7,200 | <p>We could simply apply the chain rule, to avoid some confusions we let $ C(x,t) = C(x^* + vt,t^*) = C^*(x^*,t^*)$:
$$
\frac{\partial C}{\partial x} = \frac{\partial C^*}{\partial x^{\phantom{*}}}= \frac{\partial C^*}{\partial x^*} \frac{\partial x^*}{\partial x^{\phantom{*}}} + \frac{\partial C^*}{\partial t^*} \frac... |
3,757,864 | <p>Given a diagram like this,
<a href="https://i.stack.imgur.com/Xwum0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Xwum0.png" alt="enter image description here" /></a></p>
<p>Where <span class="math-container">$O$</span> is the center and <span class="math-container">$OA = \sqrt{50}$</span>, <spa... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/IVsYw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IVsYw.png" alt="enter image description here" /></a></p>
<p>A slight variation of the solution</p>
<p>Note that <span class="math-container">$R$</span> is circumradius of <span class="math-container">$\triangl... |
68,386 | <p>I'm looking for a theorem of the form </p>
<blockquote>
<p>If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.</p>
</blockquote>
<p>My attempts to do this purely algebraically are not working, so I started looking into methods fr... | Hailong Dao | 2,083 | <p>The dimension of $R[1/v]$ is the biggest height of some prime ideal $P$ such that $v\notin P$. So, let $I_{d-1}$ be the intersection of all primes of height at least $d-1$ ($d= \dim R$), then</p>
<blockquote>
<p>$\dim R[1/v] \geq d-1$ if and only if $v\notin I_{d-1}$. </p>
</blockquote>
<p>Under a mild condition... |
3,464,247 | <p>I'm wondering if my reasoning is justified when determining if a vector is in the span of of a set of vectors.</p>
<p><span class="math-container">$$T = \{(1, 1, 0), (-1, 3, 1)\}$$</span></p>
<p>For which <span class="math-container">$a$</span> is <span class="math-container">$(a^2, a+2, 2) \in span(T)$</span></p>... | José Carlos Santos | 446,262 | <p>Actually, the answer depends upen the field that you are working with. In fact,<span class="math-container">\begin{align}\begin{bmatrix}a^2\\a+2\\2\end{bmatrix}\in\operatorname{span}\left\{\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}-1\\3\\1\end{bmatrix}\right\}&\iff\begin{vmatrix}1&-1&a^2\\1&... |
3,290,514 | <p>I need to make the navigation and guidance of a vehicle (a quadcopter) in a platform. This platform can be seen like this:</p>
<p><a href="https://i.stack.imgur.com/jeJ34.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jeJ34.png" alt="enter image description here"></a></p>
<p>where the blue dots... | Robert Israel | 8,508 | <p>One way to fit points to a circle is to do linear regression on the equation in the form <span class="math-container">$x^2 + a x + y^2 + b y + c = 0$</span>. Writing this as
<span class="math-container">$(x+a/2)^2 + (y+b/2)^2 = (a^2 + b^2)/4 - c$</span>, this corresponds to a circle of centre <span class="math-conta... |
2,785,698 | <p>Please help me go over this problem; I am a bit confused.</p>
<p>Find ${\displaystyle \frac{\mathrm d}{\mathrm dt} \int_2^{x^2}e^{x^3}\mathrm dx}$.</p>
| tien lee | 557,074 | <p>The Fundamental Theorem of Calculus states that if
$$g(x) = \int_{a}^{f(x)} h(t)~{\rm d}t$$
where $a$ is any constant, then
$$g'(x) = h(f(x)) \cdot f'(x)$$
Using this with the integral, $g(x) = g(x)$, $f(x) = x^2$, and $h(x) = e^{x^3}$.</p>
<p>So, $$\frac{\mathrm d}{\mathrm dx} \int_2^{x^2}e^{x^3}\mathrm dx=(2x)e^{... |
4,328,117 | <p>I have this lambda expression</p>
<p><span class="math-container">$$(\lambda xyz.xy(zx)) \;1\; 2\; 3$$</span></p>
<p>or</p>
<p><span class="math-container">$$(\lambda x. (\lambda y. (\lambda z.xy(zx))))\;1\;2\;3$$</span>
<span class="math-container">$$(\lambda y. (\lambda z.1y(z1))))\;2\;3$$</span>
<span class="math... | Marc van Leeuwen | 18,880 | <p>Syntactically, function application is a two-operand non-associative operation, where the latter means that <span class="math-container">$f(g\,x)$</span> has to be distinguished from <span class="math-container">$(f\,g) x$</span> (the function application operation conventionally has an empty operator symbol, in oth... |
463,239 | <p>Integrate $$\int{x^2(8x^3+27)^{2/3}}dx$$</p>
<p>I'm just wondering, what should I make $u$ equal to?</p>
<p>I tried to make $u=8x^3$, but it's not working. </p>
<p>Can I see a detailed answer?</p>
| Keith Afas | 85,076 | <p>You would use $u,du$ substitution</p>
<p>$$\int{x^2(8x^3+27)^{2/3}}dx$$
$$=\int{x^2(u)^{2/3}}dx$$
$$u=8x^3+27, du=24x^2 \implies \frac{du}{24}=x^2dx$$
$$\int{x^2(u)^{2/3}}dx$$
$$=\int{(u)^{2/3}x^2}dx$$
$$=\int{(u)^{2/3}}\frac{du}{24}$$
$$=\frac{1}{24}\int{(u)^{2/3}}du$$
$$=\frac{1}{24}\int{(u)^{2/3}}du$$
$$=\frac{1... |
1,380,697 | <p>I am currently an undergraduate and thinking about applying to graduate school for math. The problem is that I don't know what field I want to go. Taking graduate classes even more confuse me because the more I learn the less I know what specifically I want to do. My question is to where to find an information about... | Thomas | 26,188 | <p>It is hard to get a good overview of all of mathematics. The best really is to <strong>take a good broad variety of classes</strong>. This you usually do (to some extent) the first couple of years in graduate school. Here you will learn the basic language of the main areas of mathematics. Here you will be get to <st... |
3,681,916 | <p>Given a matrix A, Does
<span class="math-container">$$\lim_{n \to \infty}A^n=0$$</span>
Imply that(by lower then I mean that every number in the lower matrix is closer or the same distance to 0 then it’s counterpart in the bigger one)
<span class="math-container">$$\text{if }a>b>0 \text{ then }A^a<A^b$$</sp... | Clement Yung | 620,517 | <p>This is false. Consider:
<span class="math-container">$$ A :=
\begin{pmatrix}
0 & 0.5 \\
0.5 & 0
\end{pmatrix}
$$</span>
Then the sequence of matrix is as follows:
<span class="math-container">\begin{align*}
A^2 =
\begin{pmatrix}
0.25 & 0 \\
0 & 0.25
\end{pmatrix}, \;
A^3 =
\begin{pmatrix}
0 & 0.... |
1,910,109 | <p>$$\int \frac{1}{\sqrt{x} (1 - 3\sqrt{x})}$$</p>
<p>I tried with the substitution $u = 1-3\sqrt{x}$</p>
<p>I am confused with how to finish this problem I know I am supposed to substitute $u$ and $\text{d}u$ in but I am not sure how to finish it.</p>
| Enrico M. | 266,764 | <p>Let's set the problem, then you may proceed by yourself.</p>
<p>Performing the substitution</p>
<p>$$u = 1 - 3\sqrt{x} ~~~ \text{and so} ~~~ \sqrt{x} = -\frac{u-1}{3}$$</p>
<p>Hence</p>
<p>$$\text{d}u = -\frac{3}{2\sqrt{x}} = -\frac{3}{2}\frac{-3}{u-1}\ \text{d}x$$</p>
<p>$$\text{d}u = \frac{9}{2(u-1)}\ \text{d... |
3,931,831 | <p>For the scenario given below, I am confused about if the samples are dependent or independent since the scenario does not mention anything about the samples being paired/related or vice versa.</p>
<p>I am aware if terms such as paired, repeated measurements, within-subject effects, matched pairs, and pretest/posttes... | PierreCarre | 639,238 | <p>In this case, it is really not a great idea to use Lagrange multipliers. We can write <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span> (or vice-versa) using the restriction and reduce this question to a one variable optimisation problem. Substituting <span class="math-conta... |
1,832,812 | <blockquote>
<p>Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+$.</p>
</blockquote>
<p>Suppose $a \in M$. Then so are $4ak$ and $\lfloor \sqrt{4ak} \rfloor$ for every positive integer ... | Ethan Alwaise | 221,420 | <p>Use the fact that $\lfloor \sqrt{x} \rfloor \in M$ for every $x \in M$ to show that $1 \in M$, hence $M$ contains every multiple of $4$. To finish, use the fact that there is always a multiple of $4$ between any two consecutive squares. </p>
|
1,201,955 | <blockquote>
<p><em>Question</em>: If $f(n)$ is $O(g(n))$ and $g(n)$ is $O(f(n))$, is $f(n) = g(n)$?</p>
</blockquote>
<p>I'm studying for a discrete mathematics test, and I'm not sure if this is true or false. Since Big-OH ignores constant multiples, wouldn't $f(n) = c\, g(n)$?</p>
| marty cohen | 13,079 | <p>Consider
$f(n) = an^2$
and
$g(n) = bn^2$
where $a$ and $b$ are
distinct positive reals.</p>
<p>Both
$f(n) = O(g(n))$
and
$g(n) = O(f(n))$
are true,
but $f(n) \ne g(n)$
for $n > 0$.</p>
<p>This is often written as
$f(n) = \Theta(g(n))$.</p>
|
2,170,832 | <p>What is $\mathbb{Z}^*_p$ ?</p>
<p>I think $\mathbb{Z}_p$ is the ring of p-adic integers but not sure what the * represents.</p>
| Ege Erdil | 326,053 | <p>It's the group of $ p $-adic units.</p>
|
2,170,832 | <p>What is $\mathbb{Z}^*_p$ ?</p>
<p>I think $\mathbb{Z}_p$ is the ring of p-adic integers but not sure what the * represents.</p>
| Dietrich Burde | 83,966 | <p>The star usually refers to the group of units of a ring $R$. Another notation is $U(R)$. </p>
|
456,826 | <p>I need to find the derivative of this function. I know I need to separate the integrals into two and use the chain rule but I am stuck.</p>
<p>$$y=\int_\sqrt{x}^{x^3}\sqrt{t}\sin t~dt~.$$</p>
<p>Thanks in advance</p>
| Community | -1 | <p><strong>Hint</strong></p>
<p>By the chain rule we prove easly:</p>
<p>If
$$F(x)=\int_{u(x)}^{v(x)}f(t)dt$$
then
$$F'(x)=f(v(x))v'(x)-f(u(x))u'(x)$$</p>
|
3,989,878 | <p>I can't solve this problem. I tried to find <span class="math-container">$\tan x$</span> directly by solving cubic equations but I failed.</p>
<p>The problem is to find <span class="math-container">$\tan x\cot 2x$</span> given that
<span class="math-container">$$\tan x+ \tan 2x=\frac{2}{\sqrt{3}}, \>\>\>\&g... | Raffaele | 83,382 | <p><span class="math-container">$$\frac{2 \tan x}{1-\tan ^2 x}+\tan x=^*\frac{2}{\sqrt{3}}$$</span>.
<span class="math-container">$$\tan x=-1.45424, \tan x=0.35178, \tan x =2.25716$$</span>
<span class="math-container">$L=\tan x \cot 2x$</span> can be written<span class="math-container">$^{**}$</span> as <span class="m... |
2,400,654 | <p>I am told that the statement "any closed set has a point on its boundary" is false, yet I don't know how to disprove it. In fact, I think it is true. </p>
<p>Suppose we have [a,b], a closed set. Then, the boundary would be {a,b}, both of which are the elements of the set. So, there we have a closed set that has poi... | Michael Hardy | 11,667 | <p>On the "pro" side I would mention two facts:</p>
<ul>
<li><p>For $\alpha>0,$ let $$f_\alpha(x) = \frac 1 {\Gamma(\alpha)} x^{\alpha-1} e^{-x}.$$ Then for $\alpha,\beta>0$ we have the convolution $$f_\alpha * f_\beta = f_{\alpha+\beta}.$$ (And recall that the convolution of two probability density functions is... |
3,085,591 | <p>I'm trying to prepare for an exam and came across the following question:</p>
<blockquote>
<p>Given <span class="math-container">$n \times n$</span> matrix <em>A</em>, let <em>U</em> represent row space and <em>W</em>
represent column space. </p>
<p>A) Prove: <span class="math-container">$W \subseteq$</sp... | jmerry | 619,637 | <p>Hint: compare the dimensions of <span class="math-container">$W$</span> and <span class="math-container">$U^{\perp}$</span>, given that <span class="math-container">$\dim(U) >\frac n2$</span>. We can't include a vector space inside one of smaller dimension...</p>
|
3,292,918 | <p>Let <span class="math-container">$X$</span> be a Banach space, and denote by <span class="math-container">$B_r (x)$</span> the closed ball of radius <span class="math-container">$r > 0$</span>
around <span class="math-container">$x \in X$</span>. Furthermore, let <span class="math-container">$A \subset X$</span> ... | Selrach Dunbar | 278,932 | <p>Unfortunately, the infimum above is not actually a minimum. To see this let <span class="math-container">$A$</span> be any two point set in <span class="math-container">$\mathbb{R}^2$</span>. Then <span class="math-container">$r_0=0$</span> whenever <span class="math-container">$N \geq 2$</span>.</p>
|
2,693,143 | <p>For example, to convert $0.25$ to binary. Using this algorithm it gives the correct result $0.01$:</p>
<blockquote>
<ol>
<li>Multiply by two</li>
<li>take decimal as the digit</li>
<li>take the fraction as the starting point for the next step</li>
<li>repeat until you either get to 0 or a periodic number<... | fleablood | 280,126 | <p>Cosider that $0 < x < 1$ and in base $10$, $x = 0.abcde....$ and in base $2$ $x = 0.uvwz......$ and let $2x = h.ijklmn.....$</p>
<p>Now $x \ge \frac 12 \iff a \ge 5\iff 2x \ge 1 \iff u = 1 \iff h = 1$.</p>
<p>And $x < \frac 12 \iff a < 5\iff 2x < 1 \iff u= 0 \iff h=0$.</p>
<p>So $u = h$ and that ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.