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 |
|---|---|---|---|---|
57,281 | <p><strong>Bug introduced in 10.0 and fixed in 10.3</strong></p>
<hr>
<p>I'm having trouble calculating the median of a <code>Dataset[]</code> in <em>Mathematica</em> 10.</p>
<p>The situation is as follows. Consider a dataset that was defined as follows:</p>
<pre><code>dataset = Dataset[{<|"a"->1,"b"->2|&g... | Community | -1 | <p>Working since version 10.3:</p>
<pre><code>dataset = Dataset[{<|"a" -> 1, "b" -> 2|>, <|"a" -> 3, "b" -> 4|>}];
med = dataset[Median, {"a", "b"}]
</code></pre>
<p><a href="https://i.stack.imgur.com/cBvSQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cBvSQ.png" alt="ent... |
136,067 | <p>Assume $f(x)>0$ defined in $[a,b]$, and for a certain $L>0$, $f(x)$ satisfies the Lipschitz condition $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$.</p>
<p>Assume that for $a\leq c\leq d\leq b$,$$\int_c^d \frac{1}{f(x)}dx=\alpha,\int_a^b\frac{1}{f(x)}dx=\beta$$Try to prove$$\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alp... | tibL | 29,405 | <p>I got something which is rather close to your result but couldn't get rid of an additional term. I'm hoping someone will find the development useful in order to give a complete answer. The question reminded me somehow of the proof of Gronwall's inequality and my answer is based on that.</p>
<p>Let $h(t)=\int_a^t f(... |
2,601,412 | <p>"A game is played by tossing an unfair coin ($P(head) = p$) until $A$ heads or $A$ tails (not necessarily consecutive) are observed. What is the expected number of tosses in one game?"</p>
<p>My approach is the following:</p>
<p>Let's represent represent a head by $H$ and a tail by $T$, and call $H_n$ the event "t... | gar | 138,850 | <p>A closed form may not exist, but we can write it as summations, which can be easily evaluated by computer algera systems</p>
<p>1.
\begin{align*}
f(a,0) &= p^a \\
f(0,b) &= (1-p)^b \\
f(a,b) &= p\cdot f(a-1,b) + (1-p)\cdot f(a,b-1) \\
E(A,p) &= A\times \sum_{i=0}^{A-1} f(A,i) + f(i,A)
\end{a... |
223,631 | <p>I'm using NeumannValue boundary conditions for a 3d FEA using NDSolveValue. In one area I have positive flux and in another area i have negative flux. In theory these should balance out (I set the flux inversely proportional to their relative areas) to a net flux of 0 but because of mesh and numerical inaccuracies... | Tim Laska | 61,809 | <h1>Update (Steady-State Solution)</h1>
<p>I think the fundamental issue is that you are over constraining your system. Whether you are solving the "heat equation" or not, your operator has the same form of the heat equation as shown below:</p>
<p><span class="math-container">$$\rho {{\hat C}_p}\frac{{\parti... |
176,340 | <p>I am running an iterative routine that I want to export to a file while each iteration is computed, instead of storing everything in memory and then exporting to a file. </p>
<p>My solution is to write to an "m" file that saves the values in the usual array format that mathematica understands (e.g. {{2,1},{3,1}} fo... | rhermans | 10,397 | <p>Something like this?</p>
<p>The strategy is to pre-generate the delimiters and the coordinates to then have a single loop that writes the value and the next delimiter, whatever that is in each iteration and for whatever the number of dimensions.</p>
<pre><code>Module[
{
sm = 3,
rm = 3,
stream = OpenWrite["t... |
1,005,576 | <p>How can I write this term in a compact form where $a$ only appears once on the RHS (in particular without cases)?</p>
<p>$T(a) =
\begin{cases}
a^2 &,\text{ if $a \leq 0$}\\
2a^2 &,\text{ if $a > 0$}\\
\end{cases}$</p>
<p>I have already thought about $T(a) = \max\{\sqrt{2}a,|a|\}^2$ or $T(a)... | matheburg | 155,537 | <p>After discussing the problem with a brilliant friend we came up with the following solution:</p>
<p>$T(a) = \left[\Re\left((\sqrt[4]{2}-i)\sqrt{a}\right)\right]^4$</p>
<p>However, I am still up for further suggestions!</p>
|
2,032,711 | <p>In a triangle $ABC$, if $\sin A+\sin B+\sin C\leq1$,then prove that $$\min(A+B,B+C,C+A)<\pi/6$$
where $A,B,C$ are angles of the triangle in radians.</p>
<p>if we assume $A>B>C$,then $\sum \sin A\leq 3 \sin A$,and $ A\geq \frac{A+B+C}{3}=\pi/3$.also $\sum \sin A\geq 3\sin C$ and $ C\leq \frac{A+B+C}{3}=\pi... | dezdichado | 152,744 | <p>Since you assumed $A\geq B\geq C$, it must be that $\dfrac{A}{2}+C\leq\dfrac{\pi}{2}.$ Hence, $\sin\tfrac{A}{2}<\sin(\tfrac{A}{2}+C) = \cos(\tfrac{B-C}{2})$.
Finally, $$1\geq \sin A+\sin B+\sin C = \sin A+2\sin\tfrac{B+C}{2}\cos\tfrac{B-C}{2} = 2\cos\tfrac{A}{2}\big(\sin\tfrac{A}{2}+\cos\tfrac{B-C}{2}\big)>4\... |
22,101 | <p>The general rule used in LaTeX doesn't work: for example, typing <code>M\"{o}bius</code> and <code>Cram\'{e}r</code> doesn't give the desired outputs.</p>
| Avatar | 186,146 | <p>If you use the standard implementation of Mathjax, the <code>M\"{o}bius</code> will not render. </p>
<p>Workaround: </p>
<pre><code>\ddot{o}
</code></pre>
<p>will give: </p>
<p>$$ \ddot{o} $$</p>
<p>Probably this is helpful for some.</p>
<p>Another workaround is to specify another font for the text in Mathjax:... |
3,371,302 | <p>trying to find all algebraic expressions for <span class="math-container">${i}^{1/4}$</span>.</p>
<p>Using. Le Moivre formula , I managed to get this : </p>
<blockquote>
<p><span class="math-container">${i}^{1/4}=\cos(\frac{\pi}{8})+i \sin(\frac{\pi}{8})=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}} + i \sqrt{\frac{1-... | Allawonder | 145,126 | <p>The fourth roots are spaced on a circle, equally partitioning it. Thus, if you know one, you can find the rest by rotating by <span class="math-container">$π/2,$</span> or (which is the same thing) by multiplying by <span class="math-container">$i.$</span></p>
<p>Thus, since one of them as you found is <span class=... |
89,197 | <p>I am working on a problem where I have to generate a table of components while each component of the table has 18 entries. Six of the indices among 18 run from 0 to 1 while the other 12 can take values between 0 to 3. After doing that I have to select some of the entries which follow a certain criterion (sum of all ... | m_goldberg | 3,066 | <p>I don't know if this will save you sufficient memory, but it will certainly cut down your memory use.</p>
<pre><code>$HistoryLength = 0;
list1 =
Flatten[
Table[{i, j, k, l, m, n, o, p, q, r, s, u, v, x, y, z, a, b},
{i, 0, 1}, {j, 0, 3}, {k, 0, 3}, {l, 0, 1}, {m, 0, 3}, {n, 0, 3}, {o, 0, 1},
... |
89,197 | <p>I am working on a problem where I have to generate a table of components while each component of the table has 18 entries. Six of the indices among 18 run from 0 to 1 while the other 12 can take values between 0 to 3. After doing that I have to select some of the entries which follow a certain criterion (sum of all ... | ciao | 11,467 | <p>I think the comment solution will serve you well:</p>
<pre><code>p1 = Join @@ Permutations /@ IntegerPartitions[3, {18}, Range[0, 3]];
result = Cases[p1, Alternatives @@@ Range[0, {1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3}]];
</code></pre>
<p>Testing this (on my loungebook, so I limited both yours and ... |
1,766,264 | <p>A store sells 8 kinds of candy. How many ways can you pick out 15 candies total to throw unordered into a bag and take home.</p>
<p>here 15 candies..
so we choose 8 from out of 15 is ..=$^{15}C_8$ is i am right</p>
| mathreadler | 213,607 | <p>Something to expand a bit on Andrés answer ( <strong>may be more than you need</strong>, but could maybe be interesting if you are curious. Consider the expression.
$$(x_1+\cdots+x_8)^{15}$$ each time we pick a term in the factor $x_k$, that symbolizes picking candy type $k$. So the possible candy configurations wil... |
1,766,264 | <p>A store sells 8 kinds of candy. How many ways can you pick out 15 candies total to throw unordered into a bag and take home.</p>
<p>here 15 candies..
so we choose 8 from out of 15 is ..=$^{15}C_8$ is i am right</p>
| N. F. Taussig | 173,070 | <p>The number $\binom{15}{8}$ represents the number of ways of making an unordered selection of eight objects from a set of $15$ distinct objects. </p>
<p>In this problem, we are instead selecting $15$ pieces of candy from eight different types of candy. What matters is how many candies of each type we choose. If $... |
267,706 | <p>I'm making an animation of a <a href="https://en.wikipedia.org/wiki/Reuleaux_triangle" rel="nofollow noreferrer">Reuleaux triangle</a> rolling on a straight line like this
<a href="https://i.stack.imgur.com/m0IMm.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/m0IMm.gif" alt="rolling Reuleaux tria... | Adam | 74,641 | <p>Daniel Huber's code seems to let the points and edges slide a little. In order to ensure no slippage, you need to use the perimeter of the shape: each arc has length <span class="math-container">$\pi/3$</span>, so the distance between successive vertex rotations should be <span class="math-container">$\pi/3$</span>... |
1,865,364 | <p>After having seen a lengthy and painful calculation showing
$\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}3, \sqrt[\leftroot{-2}\uproot{2}3]{2}]/\mathbb Q)\cong S_3$, I'm wondering whether there's a slick proof $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}p, \sqrt[\leftroot{-2}\uproot{2}p]{2}]/\mathbb Q)\cong S_p$ fo... | M. Van | 337,283 | <p>Your statement does not hold. Let $\zeta$ be some $p$-th root of unity. Remember that the order of the galois group $\text{Gal} \mathbb{Q}(\zeta, \sqrt[p]{2})$ is the degree of the extension $\mathbb{Q}(\zeta, \sqrt[p]{2})/ \mathbb{Q}$. Now $[\mathbb{Q}(\zeta) : \mathbb{Q}]=p-1$ and $[\mathbb{Q}(\sqrt[p]{2}) : \math... |
3,660,652 | <p>To which of the seventeen standard quadrics (<a href="https://mathworld.wolfram.com/QuadraticSurface.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/QuadraticSurface.html</a>) do these two equations reduce?
<span class="math-container">\begin{equation}
Q_1^2+3 Q_2 Q_1+\left(3
Q_2+Q_3\right){}^2 = 3 ... | fleablood | 280,126 | <p><span class="math-container">$x = \frac{781 + 256 (3d-1)}{81}$</span> means that <span class="math-container">$x = \frac{781 + 256 (3d-1)}{81}$</span> is an integer and that <span class="math-container">$81$</span> divides into <span class="math-container">$781+256(3d-1)$</span> evenly.</p>
<p><span class="math-con... |
319,725 | <p>I am trying to prove the following inequality concerning the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="noreferrer">Beta Function</a>:
<span class="math-container">$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$</span>
where as usual <span class="math-c... | esg | 48,831 | <p>One can also use Jensen's inequality. Let (for <span class="math-container">$\sigma>0$</span>) <span class="math-container">$G_\sigma$</span> denote a random variable with <span class="math-container">$\Gamma(1,\sigma)$</span>-distribution, i.e. having Lebesgue density
<span class="math-container">$$f_\sigma(t)... |
4,351,504 | <p>A question from Herstein's Abstract Algebra book goes-</p>
<blockquote>
<p>Let <span class="math-container">$(R,+,\cdot)$</span> be a ring with unit element. Using its elements we define a ring <span class="math-container">$(\tilde R,\oplus,\odot)$</span> by defining <span class="math-container">$a\oplus b = a + b +... | Svyatoslav | 869,237 | <p>We can also try to dig a bit deeper. Knowing that <a href="https://www.google.com/search?q=erf%20laplace%20transform&rlz=1C1GCEU_ruRU866RU868&oq=erf%20laplace%20transform&aqs=chrome..69i57.11914j0j15&sourceid=chrome&ie=UTF-8" rel="nofollow noreferrer">Laplace Transform</a> of <span class="math-co... |
2,360,523 | <blockquote>
<p>Let $a, b, c$ be positive real numbers such that $a+b+c = 1$. Prove that
$$ \displaystyle\sum_{cyc}\frac{ab}{\sqrt{ab+bc}} \leq \frac{1}{\sqrt{2}}$$</p>
</blockquote>
<p>My attempted work :</p>
<p>By C-S, $$ (ab+ac)(1+1) \geq (\sqrt{ab}+\sqrt{bc})^2$$</p>
<p>$$\sqrt{2} \sqrt{ab+bc} \geq \sqrt{ab}... | Michael Rozenberg | 190,319 | <p>Your solution is wrong because
$$ 2 \displaystyle\sum_{cyc} \frac{ ab}{ \sqrt{ab}+\sqrt{bc}} \neq \displaystyle\sum_{cyc} \frac{ ab}{ \sqrt{ab}+\sqrt{bc}} + \displaystyle\sum_{cyc}\frac{ bc}{ \sqrt{ab}+\sqrt{bc}} $$</p>
<p>My proof:</p>
<p>By C-S
$$\left(\sum_{cyc}\sqrt{\frac{a^2b}{a+c}}\right)^2\leq(ab+ac+bc)\su... |
2,360,523 | <blockquote>
<p>Let $a, b, c$ be positive real numbers such that $a+b+c = 1$. Prove that
$$ \displaystyle\sum_{cyc}\frac{ab}{\sqrt{ab+bc}} \leq \frac{1}{\sqrt{2}}$$</p>
</blockquote>
<p>My attempted work :</p>
<p>By C-S, $$ (ab+ac)(1+1) \geq (\sqrt{ab}+\sqrt{bc})^2$$</p>
<p>$$\sqrt{2} \sqrt{ab+bc} \geq \sqrt{ab}... | River Li | 584,414 | <p>Some years ago, I came up with a proof.</p>
<p><strong>Proof</strong>: By using the Cauchy-Schwarz inequality, we have
<span class="math-container">\begin{align}
&\frac{xy}{\sqrt{xy+yz}}+\frac{yz}{\sqrt{yz+zx}}+\frac{zx}{\sqrt{zx+xy}}\\
\le \ & \sqrt{(xy+yz+zx)\Big(\frac{xy}{xy+yz}+\frac{yz}{yz+zx}+\frac{zx... |
332,927 | <p>there are two bowls with black olives in one and green in the other. A boy takes 20 green olives and puts in the black olive bowl, mixes the black olive bowl, takes 20 olives and puts it in the green olive bowl. The question is -</p>
<p>Are there more green olives in the black olive bowl or black olive in the green... | Philip C | 12,160 | <p>Suppose you start with $B$ black olives in one bowl and $G$ green olives in the other.</p>
<p>After the first transfer, we have $B$ black olives and $20$ green olives in one bowl and $G-20$ green olives in the other.</p>
<p>For the second transfer, suppose the boy picks $X$ black olives and $20-X$ green olives (fo... |
152,295 | <p>What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?</p>
| Carlo Beenakker | 11,260 | <p><A HREF="http://arxiv.org/abs/hep-th/9706033" rel="nofollow noreferrer">Picture changing operators in supergeometry and superstring theory</A>, Alexander Belopolsky (1997).</p>
<p><IMG SRC="https://ilorentz.org/beenakker/MO/picture_change.png"></p>
|
3,196,238 | <p>Let <span class="math-container">$ A = \left\{ (x,y) \in \mathbb{R^2} \mid y= \sin ( \frac{1}{x}) , \ 0 < x \leq 1 \right\}$</span> . Find <span class="math-container">$\operatorname{Cl} A$</span> in topological space <span class="math-container">$\mathbb{R^2}$</span> with dictionary order topology.</p>
<p>I ... | Cameron Buie | 28,900 | <p>You're quite right. To prove it, I would take an arbitrary point not in <span class="math-container">$A,$</span> and find an open interval around it containing no points of <span class="math-container">$A.$</span> This shows that the complement of <span class="math-container">$A$</span> is open, so that <span class=... |
3,520,327 | <p>Currently in Calculus II and I was introduced to hyperbolic trigonometric functions and it threw me for a loop. I’m really confused on their MEANING... and what they represent. I can use the formulas for them easily but it doesn’t actually make sense to me. Can someone please help me out? Are there any good books yo... | Dan Christensen | 3,515 | <p>In classical logic, <span class="math-container">$p \implies q$</span> means only that it is false that both <span class="math-container">$p$</span> is true and <span class="math-container">$q$</span> is false. </p>
<blockquote>
<p><span class="math-container">$p \implies q \space \space \equiv \space\space \neg ... |
9,345 | <p>On meta.tex.sx, I've asked a question about a class of questions that might get asked over there (and have been) that are (i) ostensibly about maths usage, but (ii) might best be served by an answer that is primarily about how to handle the notation in Latex (See <a href="https://tex.meta.stackexchange.com/questions... | Tom Oldfield | 45,760 | <p>If your question is overlapping the two disciplines, it may be best to split it into two distinct parts, and post these parts separately in the respective exchanges. Saying this, it is my opinion that the example in your meta post on the TeX exchange belonged solely here and not there, since no part of the question ... |
250,364 | <blockquote>
<p><strong>Problem</strong> Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous.</p>
</blockquote>
<p>My idea is to consider $|x - y| < \delta$, then show that
$$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})|
= \bigg|\log\bigg(\dfrac{1 + \sqrt{1+x^2}}{1 + \sqrt{1+y^2}}\bigg)\bigg| <... | lhf | 589 | <p>The derivative of $f(x)=\log(1 + \sqrt{1+x^2})$ is $\frac{x}{1 + x^2 + \sqrt{1 + x^2}}$, which is bounded in the whole real line since it is continuous and tends to $0$ as $x\to\pm\infty$. By the Mean Value Theorem, $f$ is Lipschitz and so uniformly continuous.</p>
|
1,281,967 | <p>This is a dumb question I know.</p>
<p>If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$.</p>
<p>But multiplication is not commutative in matrix math. Would it be correct to state that I can solve for $A^{-1}Ax = A^{-1}b \implies x =... | the.polo | 202,381 | <p>Yes, if the matrix is invertible, this is correct and the equation has the unique solution $x=A^{-1}b$.</p>
<p>Here is the <a href="http://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem" rel="nofollow">list</a> of properties that make a matrix invertible.</p>
|
1,082,390 | <p>$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} - \sqrt{4x^2+x}\ \right)$$</p>
<p>I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\di... | lab bhattacharjee | 33,337 | <p>Set $h=\dfrac1x$ </p>
<p>$$4x^2+ax=\frac{4+ah}{h^2}\implies\sqrt{4x^2+ax}=\frac{\sqrt{4+ah}}{\sqrt{h^2}}$$</p>
<p>Now as $h\to0^+,h>0\implies\sqrt{h^2}=|h|=h$</p>
<p>$$\implies\lim_{x \to \infty} (\sqrt{4x^2+5x} - \sqrt{4x^2+x})=\lim_{h\to0^+}\frac{\sqrt{4+5h}-\sqrt{4+h}}h$$</p>
<p>$$=\lim_{h\to0^+}\frac{4+5h... |
1,082,390 | <p>$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} - \sqrt{4x^2+x}\ \right)$$</p>
<p>I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\di... | Claude Leibovici | 82,404 | <p>Since you already received answers, let me show you another approach you could use. $$A=\sqrt{4x^2+5x} - \sqrt{4x^2+x}=2x \sqrt{1+\frac{5}{4x}}-2x \sqrt{1+\frac{1}{4x}}=2x \Big(\sqrt{1+\frac{5}{4x}}-\sqrt{1+\frac{1}{4x}}\Big)$$ Now, you may be already know that, when $y$ is small compared to $1$ $$\sqrt{1+y}=1+\frac... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Mr. Brooks | 162,538 | <p>You can also use the <em>least</em> significant digits to get your bearings. Since $2016 \equiv 16 \pmod{100}$, if $2016$ is a perfect square, then $n$ in $n^2 = 2016$ is an integer satisfying $n \equiv 4, 6 \pmod{10}$. Clearly $n = 4$ or $6$ is too small.</p>
<p>Then, working our way up, we get $(196, 256), (576, ... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Jam | 161,490 | <p>There are well enough good answers here, but no one's suggested this method yet, so I'll add it.</p>
<h3>Method <span class="math-container">$1$</span> - (Secant Approximation)</h3>
<p>We can take two simple perfect squares that straddle <span class="math-container">$2017$</span>. We're just getting a rough estimate... |
3,393,466 | <p>I am in final year of my undergraduate in mathematics from a prestigious institute for mathematics. However a thing that I have noticed is that I seem to be slower than my classmates in reading mathematics. As in, how muchever I try, I seem to finish my works at the last moment and I rarely find any time for extra r... | RyRy the Fly Guy | 412,727 | <p>I am in the same position as you, @Deepakms. I'm always the last to finish on a math exam, the last to turn in the class assignment, etc... with that being said, I typically ace every exam, and I'm usually the one with the highest grade in the class. </p>
<p>People who invest more time in cultivating a rich underst... |
327,750 | <p>$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$</p>
<p>The results is obvious enough, but how to prove this</p>
| dtldarek | 26,306 | <p>This is a similar approach, but using different tools. It came out a bit over-formalized, but perhaps it still might be helpful to you.</p>
<hr>
<p>You want to prove</p>
<p>$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap \ldots\cap A_{n-1}^c \cap A_{n})$$
or more concisely</p>
<p>$$\bigcup_{n=1}^... |
1,074,534 | <p>How can I get started on this proof? I was thinking originally:</p>
<p>Let $ n $ be odd. (Proving by contradiction) then I dont know.</p>
| Aaron | 140,411 | <p>To get you started</p>
<p>Assume that the largest number that is divisible by 500 different numbers is $n$, then assume $n$ is not divisible by $2$. and is instead divisible by $x$, which is the smallest positive integer than $n$ can be divided by, hence $x$ must be larger than 2.</p>
<p><strong>To finish the proo... |
1,074,534 | <p>How can I get started on this proof? I was thinking originally:</p>
<p>Let $ n $ be odd. (Proving by contradiction) then I dont know.</p>
| Henry | 6,460 | <ul>
<li><p>The smallest number with at least $500$ divisors is $2^6\times 3^2 \times 5^2 \times 7 \times 11 \times 13 = 14414400$</p></li>
<li><p>The smallest number with at exactly $500$ divisors is $2^4\times 3^4 \times 5^4 \times 7 \times 11 = 62370000$</p></li>
<li><p>The smallest number with at exactly $500$ div... |
201,060 | <p>I posted this question on Math, but there has been silence there since. So, I wonder if anyone here can get any closer to the answer to my question using Mathematica. Here is the question:</p>
<p>Suppose I draw <span class="math-container">$N$</span> random variables from independent but identical uniform distribut... | JimB | 19,758 | <p><em>Mathematica</em> does make this pretty easy. The statistic of interest is the typical estimator of the median when the sample size is even. When the sample size is odd the sample median has a beta distribution:</p>
<pre><code>OrderDistribution[{UniformDistribution[{0, 1}], n}, (n + 1)/2]
(* BetaDistribution[(... |
3,068,782 | <p>The canonical basis is not a Schauder basis of the space of bounded sequences, but in some way, it uniquely determines every element in the space. Is it a basis in a weaker sense? How is it called?</p>
<p>Thanks a lot.</p>
| SmileyCraft | 439,467 | <p>A subspace of a normed vector space is closed if and only if it is weakly closed <a href="https://math.stackexchange.com/questions/449301/closed-iff-weakly-closed-subspace">Closed <span class="math-container">$\iff$</span> weakly closed subspace</a>. Hence, a set is a Schauder basis if and only if it is a "basis in ... |
2,064,284 | <blockquote>
<p>Prove that the sequence $\{y_n\}$ where $y_{n+2}=\frac{y_{n+1} +2 y_{n}}{3}$ $n\geq 1$, $0<y_1<y_2$, is convergent by using subsequencial criteria, <strong>by showing $\{y_{2n}\}$ and $\{y_{2n-1}\}$ converges to the same limit. Find the limit also</strong>.</p>
</blockquote>
<p>I can solve it b... | Asinomás | 33,907 | <p>Let $P$ be the minimum polynomial of $M$, fuppose $Q$ is a polynomial with the same degree as $M$ that annihalates $M$.</p>
<p>Write $P$ as $\alpha Q+R$ were $R$ is of degree less than $P$ (we can do this with the division algorithm).</p>
<p>Notice that $0=P(M)=\alpha Q(M)+R(M)=0+R(M)$. This implies $R$ is the zer... |
2,064,284 | <blockquote>
<p>Prove that the sequence $\{y_n\}$ where $y_{n+2}=\frac{y_{n+1} +2 y_{n}}{3}$ $n\geq 1$, $0<y_1<y_2$, is convergent by using subsequencial criteria, <strong>by showing $\{y_{2n}\}$ and $\{y_{2n-1}\}$ converges to the same limit. Find the limit also</strong>.</p>
</blockquote>
<p>I can solve it b... | Will Jagy | 10,400 | <p>There are other things that happen when the minimal polynomial and characteristic polynomial coincide; note that we demand both monic...</p>
<p>First, while there may be eigenvalues with multiplicity greater than one, nevertheless each eigenvalue occurs in a single Jordan block.</p>
<p>Second, if we call our matri... |
2,596,098 | <p>For a square matrix $A$ and identity matrix $I$, how does one prove that $$\frac{d}{dt}\det(tI-A)=\sum_{i=1}^n\det(tI-A_i)$$ Where $A_i$ is the matrix $A$ with the $i^{th}$ row and $i^{th}$ column vectors removed?</p>
| copper.hat | 27,978 | <p>Here is one way to see this:</p>
<p>Note that the map $\phi(t_1,...,t_n) = \det ( \sum_k t_k e_k e_k^T -A)$ is smooth, and if $\tau(t) = (t,....,t)$ then $f(t)=\det (tI-A) = \phi(\tau(t))$.</p>
<p>In particular, $f'(t) = \sum_k {\partial \phi(t,....,t) \over \partial t_k}$.</p>
<p>If we adopt the notation $\det B... |
853,774 | <blockquote>
<p>If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$.</p>
</blockquote>
<p>I know that I have to show $G$ is commutative, ie $a * b = b * a$</p>
<p>I have done this by first using $a^{-1}$ on the left, then $b^{-1}$ on the right, and I end up with and expres... | mwmjp | 161,182 | <p>For notational ease, let's write $ab$ in place of $a*b$. This is ultimately, an application of left and right-cancellation in a group. Namely, $$(ab)^2=a^2b^2$$ and expanding each side we see that $$abab=aabb.$$ Canceling on the left we get $bab=abb$ and now canceling on the right we have that $ba=ab$. Hence, $G$ is... |
3,407,368 | <p>Please help me to think through this.</p>
<p>Take Riemann, for example. Finding a non-trivial zero with a real part not equal to <span class="math-container">$\frac{1}{2}$</span> (i.e., a counterexample) would disprove the conjecture, and also so it to be decidable.</p>
<p>How about demonstrating that Riemann is u... | Peter | 82,961 | <p>Statements of this form (the Goldbach conjecture is another such statement) that would be proven to be true if they were proven to be undecidable in ZFC, cannot be shown to be undecidable in ZFC within ZFC.</p>
<p>The reason is that such a proof of undecidability could not work in ZFC because this would proof the s... |
3,407,368 | <p>Please help me to think through this.</p>
<p>Take Riemann, for example. Finding a non-trivial zero with a real part not equal to <span class="math-container">$\frac{1}{2}$</span> (i.e., a counterexample) would disprove the conjecture, and also so it to be decidable.</p>
<p>How about demonstrating that Riemann is u... | saulspatz | 235,128 | <p>Certainly, if the Riemann hypothesis is false, it's decidable, since there is a counter-example, as you say. It's conceivable that it's true but undecidable, since we would never get done checking zeros. This doesn't mean that the Riemann hypothesis actually is undecidable, because brute force is not the only way ... |
4,506,093 | <p><em>Please note that the following is not a duplicate:</em></p>
<p><a href="https://math.stackexchange.com/q/657931/104041">Why negating universal quantifier gives existential quantifier?</a></p>
<p><em>I am asking for a particular type of formal proof. I have added the <a href="/questions/tagged/alternative-proof" ... | peterwhy | 89,922 | <p>This is what I did a few days ago, when I was playing with the proof checker and wanted to understand how <a href="https://proof-checker.org/rules.html" rel="nofollow noreferrer">universal derivation</a> works in that tool.</p>
<p><span class="math-container">$$\begin{array}{|rll}
1 & \neg\forall x\ Px\\
\hline
... |
4,506,093 | <p><em>Please note that the following is not a duplicate:</em></p>
<p><a href="https://math.stackexchange.com/q/657931/104041">Why negating universal quantifier gives existential quantifier?</a></p>
<p><em>I am asking for a particular type of formal proof. I have added the <a href="/questions/tagged/alternative-proof" ... | Graham Kemp | 135,106 | <blockquote>
<p>That is, I can start (a subproof) by assuming ¬∀xPx. My problem is that, as far as I can see, none of the given rules allows me to go from ¬∀xPx to ¬Pa, which is what I am guessing is the next line, especially if the tableaux method is anything to go by.</p>
</blockquote>
<p>No, start <em>exactly</em> a... |
2,735,984 | <p>I tried to solve this recurrence by taking out $n+1$ as a common in the RHS, but still have $n \cdot a_n$ and $a_n$</p>
| Sungjin Kim | 67,070 | <p><strong>Hint</strong> </p>
<p>$$a_{n}=\frac{n+1}na_{n-1}+3n+3,$$
$$
na_n= (n+1)a_{n-1} + 3n(n+1)
$$</p>
<p>$$
\frac{a_n}{n+1}=\frac{a_{n-1}}n +3
$$</p>
|
1,686,568 | <p>I am learning about tensor products of modules, but there is a question which makes me very confused about it! </p>
<p>If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a balanced map (or bilinear map) $E\times F\to E\otimes F$. If some element $x\otimes y \in E\otimes F$ is $0$, then ... | Tanner Strunk | 346,324 | <p>I found this question while seeking to answer a related one. I think it's worth saying that $0\otimes n = (0\cdot 0)\otimes n = 0\cdot(0\otimes n) = 0\otimes(0\cdot 0) = 0\otimes 0 = 0\in M\otimes_A N$ where $M$ and $N$ are left $A$-modules (or right--I can never remember which is which).</p>
<p>(You may also note ... |
2,835,767 | <p>Let $V \subset L^2(\Omega)$ be a Hilbertspace and $\{V_n\}$ a sequence of subspaces such that
\begin{align*}
V_1 \subset V_2 \subset \dots \quad \text{and} \quad \overline{\bigcup_{n \in \mathbb{N}} V_n} = V \, (\text{w.r.t. } V\text{-norm} ).
\end{align*}
For some $f\in L^2(\Omega)$ we define $\phi_n = \sup_{\| v_n... | ramanujan | 235,721 | <p>A is true. Let $p_1(x) = a_1 x^2$ and $p_2(x)= a_2 x^2$ are two elements of the given set, where $a_1,a_2 \in \mathbb R$ then $p_1(x)+p_2(x)= (a_1+a_2)x^2$, which is also belongs to the same set( because $a_1+a_2 \in \mathbb R)$. So it is closed under addition. And let $k \in \mathbb R$ then $kp (x)=(k.a) x^2$, so i... |
2,655,075 | <p>How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6?</p>
<p>So I have split this problem into cases. The first case is if 6 is in the subset and the second is where 6 is not. </p>
<p>In case 1, I did 6 with 0, 1, 2, 3, 4, and 5 numbers surrounding it which yielded 1+16+36+16+1 = 70</p>
<p>My struggle... | Misha Lavrov | 383,078 | <p>For a set not containing $6$ to have a median of $6$, it must have an even number of elements, the middle two of which average to $6$. So they can be $5$ and $7$, or $4$ and $8$, or $3$ and $9$, or $2$ and $10$, or $1$ and $11$.</p>
<p>Each of these is handled identically to the case where $6$ occurs in the set and... |
206,825 | <p>Let's say i have</p>
<p>N1 = -584</p>
<p>N2 = 110</p>
<p>Z = 0.64 </p>
<p>How do i calculate from Z which value is it in range of N1..N2? Z is range from 0 to 1.</p>
| Ross Millikan | 1,827 | <p>If by $Z=0.64$ you want a number that $64\%$ of the way from $-584$ to $110,$ the expression is $-584 + 0.64(110-(-584))$</p>
|
1,849,797 | <p>Complex numbers make it easier to find real solutions of real polynomial equations. Algebraic topology makes it easier to prove theorems of (very) elementary topology (e.g. the invariance of domain theorem).</p>
<p>In that sense, what are theorems purely about rational numbers whose proofs are greatly helped by the... | goblin GONE | 42,339 | <p>Maybe this is a little trivial, but I consider the ability to rewrite $$A=\{x \in \mathbb{Q} : x^2 < 2\}$$ as $$A=\{x \in \mathbb{Q} : -\sqrt{2} <x<\sqrt{2}\}$$
to be a benefit.</p>
<p>The latter characterization makes the "structure" of this set much clearer; in particular, its suddenly clear why this set... |
2,419,116 | <p>The problem is:</p>
<p>Prove the convergence of the sequence </p>
<p>$\sqrt7,\; \sqrt{7-\sqrt7}, \; \sqrt{7-\sqrt{7+\sqrt7}},\; \sqrt{7-\sqrt{7+\sqrt{7-\sqrt7}}}$, ....</p>
<p>AND evaluate its limit.</p>
<p>If the convergen is proved, I can evaluate the limit by the recurrence relation</p>
<p>$a_{n+2} = \sqrt{7... | Simply Beautiful Art | 272,831 | <p>To prove the limit exists, show that$$a_{4n}>a_{4n+1}>2>a_{4n+3}>a_{4n+2}$$Using induction. For example,$$a_{4n}>2\implies\underbrace{a_{4n+2}=\sqrt{7-\sqrt{7+a_{4n}}}<\sqrt{7-\sqrt{7+2}}=2}_{\huge a_{4n+2}<2}$$Same with</p>
<p>$a_{4n+1}\implies a_{4n+3},\\a_{4n+2}\implies a_{4n+4},\\a_{4n+3}\i... |
1,974,114 | <p>Let R be a integral domain with a finite number of elements. Prove that R is a field.</p>
<p>Let a ∈ R \ {0}, and consider the set aR = {ar : r ∈ R}. </p>
<p>Guessing i will have to show that |aR| = R, and deduce that there exists r ∈ R such that ar = 1 but don't know what to do?</p>
| Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>If $R$ is an integral domain, multiplication by $a\ne 0$ in $R$ is an injective ring homomorphism. Now, for a map between sets with the same finite cardinality,
$$\text{injective}\iff\text{surjective}\iff\text{bijective}. $$</p>
|
462,569 | <blockquote>
<p>Consider the polynomial ring <span class="math-container">$F\left[x\right]$</span> over a field <span class="math-container">$F$</span>. Let <span class="math-container">$d$</span> and <span class="math-container">$n$</span> be two nonnegative integers.</p>
<p>Prove：<span class="math-container">$x^d-1 \... | user49685 | 49,685 | <p>You can use <em>Long Division</em> to prove that if $d$ does not divide $n$, then when dividing $x^n - 1$ by $x^d - 1$, the remainder will be $x^r - 1$. So unless $r = 0$, $x^d - 1 \not | x^n - 1$.</p>
<p>The other way round, i.e $\Leftarrow$ should be obvious.</p>
|
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Ronnie Brown | 28,586 | <p>I agree with the importance of the points mentioned by Sasha!</p>
<p>But the question is also: what is your process of studying? I found out the hard way that I learn best when I try to write out mathematics to make it as clear as possible to myself, and even as pretty as I can make it. Sometimes this has resulted ... |
2,223,577 | <p>$\mathbb{Q}[e^{\frac{2\pi i}{5}}]$ is an extension of $\mathbb{Q}$ of degree 4, since $x^4+x^3+x^2+x+1$ is the irreducible polynomial of $\theta=e^{\frac{2\pi i}{5}}$ over $\mathbb{Q}$.</p>
<p>I'm asked if there is a quadratic extension $K$ of $\mathbb{Q}$ inside $\mathbb{Q}[e^{\frac{2\pi i}{5}}]$.
I suspect that ... | Community | -1 | <p>$\theta$ is a fifth root of unity; $\mathbb{Q}(\theta) / \mathbb{Q}$ is an <em>abelian</em> extension. That is, it is a Galois extension with abelian Galois group.</p>
<p>Every abelian group $G$ of order $n$ has, for every $m \mid n$, at least one subgroup $H$ of order $m$.</p>
<p>Consequently, the extension $\mat... |
2,210,871 | <p>I'm doing some calculus homework and I got stuck on a question, but eventually figured it out on my own. My textbook doesn't have all the answers included (it only gives answers to even numbered questions for some reason). Anyways I got stuck when I needed to solve for x for this function.</p>
<p>$${\ -3x^3+8x-4{\s... | user344249 | 344,249 | <p>Most colleges offer tutoring services that are really effective in helping you learn the material and helping you to actually understand it outside of class. I would take advantage of those as much as you can because they are often free and the people tutoring really want to help you learn and understand it.</p>
|
1,903,416 | <p>Is there a function that can be bijective, with the set of natural numbers as domain and range, other than $f(n) = n$?</p>
| Hagen von Eitzen | 39,174 | <p>There are uncountably many of such maps.</p>
<p>In fact, let $A$ be any subset of $\Bbb N=\{1,2,3,\ldots\}$ such that both $A$ and $\Bbb N\setminus A$ are infinite (for example, $A$ could be the set of primes or the set of perfect squares).
Then we can define $a(n):=$ $n$th smallest element of $A$, $b(n):=$ $n$th s... |
25,172 | <p>What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?</p>
| guest troll | 19,801 | <p>There's nothing wrong with learning it rigorously, but I would recommend to learn it "non-rigorously" first and then rigorously. E.g. in the context of a "graduate" ODE course. That's because there are some very important aspects of the topic that you may miss if you only concentrate on rigor, ... |
919,040 | <p>I want to prove that a function defines a group action:</p>
<blockquote>
<p>We have group $G$ of diagonal $2\times 2$ matrices under matrix multiplication, and the set $X$ of points of the Cartesian plane, eg:</p>
<p>$G = \left\{ \begin{bmatrix} a &0\\0&b \end{bmatrix} : a,b\in \mathbb{R} - \{0\} \ri... | Martin Sleziak | 8,297 | <p>In fact, if we work with column vectors, the group action you described is just the multiplication of matrices.</p>
<p>$$g(x,y)= \begin{pmatrix}a&0\\0&b\end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax \\ by \end{pmatrix}$$</p>
<p>Now the fact that this is indeed a <a href="http://en.... |
919,040 | <p>I want to prove that a function defines a group action:</p>
<blockquote>
<p>We have group $G$ of diagonal $2\times 2$ matrices under matrix multiplication, and the set $X$ of points of the Cartesian plane, eg:</p>
<p>$G = \left\{ \begin{bmatrix} a &0\\0&b \end{bmatrix} : a,b\in \mathbb{R} - \{0\} \ri... | Pece | 73,610 | <p>You might have seen that a group action $G \times X \to X$ is actually <em>the same things</em> as a group morphism $G \to \operatorname{Bij}(X)$. Namely, for a group action $\varphi \colon G \times X \to X$, the group morphism is $\psi \colon g \mapsto \varphi(g,\cdot)$ ; conversely, any group morphism $\psi \colon... |
2,954,929 | <p>What happens when <span class="math-container">$x < -2$</span> ? Does the whole square root term just "disappear" which leaves us with 1 which is positive and thus the answer to the question is <span class="math-container">$x\le-1$</span>? Or do we have to constrain the domain of <span class="math-container">$x$<... | J.G. | 56,861 | <p>If <span class="math-container">$X$</span> is a vector with Hermitian components, <span class="math-container">$$\langle\psi |X\cdot X|\psi\rangle=\sum_i\langle\psi |X_i^2| \psi\rangle=\sum_i\langle\psi |X_i^TX_i| \psi\rangle=\sum_i\Vert X_i|\psi\rangle\Vert^2\ge 0.$$</span></p>
|
3,678,417 | <p>I understand:
<span class="math-container">$$\sum\limits^n_{i=1} i = \frac{n(n+1)}{2}$$</span>
what happens when we restrict the range such that:
<span class="math-container">$$\sum\limits^n_{i=n/2} i = ??$$</span></p>
<p>Originally I thought we might just have <span class="math-container">$\frac{n(n+1)}{2}/2$</spa... | Alex | 38,873 | <p>Asymptotic solution: the difference, let's call it <span class="math-container">$S_d = S_1 - S_2$</span> can be obtained by taking the largest term in each sum, i.e. <span class="math-container">$\frac{n(n+1)}{2}$</span> and <span class="math-container">$\frac{n(n+2)}{8}:$</span>
<span class="math-container">$$
S_d ... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | Gerry Myerson | 8,269 | <p><span class="math-container">$2^{2i}z$</span> <strong>does</strong> cause a stretching of <span class="math-container">$z$</span>. It causes a stretching by the magnitude of <span class="math-container">$2^{2i}$</span>. And the magnitude of <span class="math-container">$2^{2i}$</span> is <strong>one</strong>. So, it... |
2,148,187 | <p>I am given a charge of $Q(t)$ on the capacitor of an LRC circuit with a differential equation</p>
<p>$Q''+2Q'+5Q=3\sin(\omega t)-4\cos(\omega t)$ with the initial conditions $Q(0)=Q'(0)=0$</p>
<p>$\omega >0$ which is constant and $t$ is time. I am then asked find the steady state and transient parts of the solu... | BobaFret | 43,760 | <p>You're right so far! Your $Q(t)$ is correct.</p>
<p>Determining the values of $c_1$ and $c_2$ isn't too bad compared to what you've done so far. You should get:</p>
<p>$$c_1 = \dfrac{1}{2} \dfrac{3\omega^3 + 4 \omega^2 - 9\omega + 20}{\omega^4 - 6\omega^2 + 25}$$</p>
<p>$$c_2 = - \dfrac{2( 2\omega^2 - 3\omega -10... |
2,148,187 | <p>I am given a charge of $Q(t)$ on the capacitor of an LRC circuit with a differential equation</p>
<p>$Q''+2Q'+5Q=3\sin(\omega t)-4\cos(\omega t)$ with the initial conditions $Q(0)=Q'(0)=0$</p>
<p>$\omega >0$ which is constant and $t$ is time. I am then asked find the steady state and transient parts of the solu... | Jan Eerland | 226,665 | <p>Another way of solving is to use Laplace transform, we have that:</p>
<p>$$\mathcal{Q}''\left(t\right)+2\cdot\mathcal{Q}'\left(t\right)+5\cdot\mathcal{Q}\left(t\right)=3\cdot\sin\left(\omega t\right)-4\cdot\cos\left(\omega t\right)\tag1$$</p>
<p>Now, in order to take the Laplace transform of both sides, use this:<... |
388,523 | <p>I have this question:</p>
<p>Evaluate $\int r . dS$ over the surface of a sphere, radius a, centred at the origin. </p>
<p>I'm not really sure what '$r$' is supposed to be? I would guess a position vector? If so, I would have $r . dS$ as $(asin\theta cos\phi, a sin\theta sin\phi, acos\theta) . (a^2sin\theta d\thet... | Andrés E. Caicedo | 462 | <p>There is a hierarchy of fast-growing functions $f_\alpha:\mathbb N\to\mathbb N$ indexed by ordinals $\alpha<\epsilon_0$, where $\epsilon_0$ is the first ordinal fixed point of the ordinal exponentiation map $\tau\mapsto\omega^\tau$, that is,
$$ \epsilon_0=\omega^{\omega^{\dots}}. $$
(Note that this is a countabl... |
3,232,296 | <ol>
<li><p>For , ∈ ℝ, we have ‖−‖≤‖+‖. </p></li>
<li><p>The dot product of two vectors is a vector. </p></li>
<li><p>For ,∈ℝ, we have ‖−‖≤‖‖+‖‖. </p></li>
<li><p>A homogeneous system of linear equations with more equations than variables will always have at least one parameter in its solution. </p></li>
<li><p>Given a... | Paulo Mourão | 673,659 | <p><span class="math-container">$1$</span> and <span class="math-container">$2$</span> are both right.</p>
<p><span class="math-container">$3$</span> is wrong. The triangle inequality actually implies <span class="math-container">$3$</span>:</p>
<p><span class="math-container">$$||u-v||\leq ||u||+||-v||=||u||+||v||$$... |
3,046,205 | <p>I am trying to figure out the steps between these equal expressions in order to get a more general understanding of product sequences:
<span class="math-container">$$\prod_{k=0}^{n}\left(3n-k\right) + \prod_{k=n}^{2n-3}\left(2n-k\right) = \prod_{j=2n}^{3n}j + \prod_{j=3}^{n}j =\frac{(3n)!}{(2n-1)!}+\frac{n!}{2}$$</s... | Andreas Caranti | 58,401 | <p>Note first that periodic here means there are <span class="math-container">$K$</span>, <span class="math-container">$P > 0$</span> such that <span class="math-container">$a^{i} = a^{i+P}$</span> for <span class="math-container">$i > K$</span>.</p>
<p>One begins with showing that there are <span class="math-co... |
4,156,482 | <blockquote>
<p>Can every continuous function <span class="math-container">$f(x)$</span> from <span class="math-container">$\mathbb{R}\to \mathbb{R}$</span> be continuously "transformed" into a differentiable function?</p>
</blockquote>
<p>More precisely is there always a continuous (non constant) <span cla... | Frank | 460,691 | <p><em>Edit 1.</em> I realized my proof has a mistake. Because I use the inverse function theorem on <span class="math-container">$g \circ f$</span>, my answer only checks out if <span class="math-container">$g \circ f$</span> is required to be <em>continuously differentiable</em>.</p>
<hr />
<p><em>Original answer.</e... |
4,156,482 | <blockquote>
<p>Can every continuous function <span class="math-container">$f(x)$</span> from <span class="math-container">$\mathbb{R}\to \mathbb{R}$</span> be continuously "transformed" into a differentiable function?</p>
</blockquote>
<p>More precisely is there always a continuous (non constant) <span cla... | andrew bruckner | 946,388 | <p>Gillis, Note on a conjecture of Erdos, Quart. J. of Math.,Oxford 10, (1939),151-154, has an example of a continuous function <span class="math-container">$f$</span>, all of whose level sets are Cantor sets. Unless <span class="math-container">$g$</span> is a constant function, it seems <span class="math-container">$... |
3,604,388 | <p>Let <span class="math-container">$P_n$</span> be the statement that <span class="math-container">$\dfrac{d^{2n}}{dx^{2n}}(x^2-1)^n = (2n)!$</span> </p>
<p>Base case: n = 0, <span class="math-container">$\dfrac{d^0}{dx^0}(x^2-1)^0 = 1 = 0!$</span></p>
<p>Assume <span class="math-container">$P_m = \dfrac{d^m}{dx^m}... | Calvin Lin | 54,563 | <p>You've made several mistakes. </p>
<p><strong>Hint:</strong> What does the product rule say <span class="math-container">$\frac{d}{dx} f(x) g(x)$</span> is equal to?<br>
Now set <span class="math-container">$ f(x) = x^2 -1, g(x) = (x^2 -1)^m$</span>. </p>
|
3,834,796 | <p>I found a really interesting question which is as follows:
Prove that the value of
<span class="math-container">$$\sum^{7}_{k=0}[({7\choose k}/{14\choose k})*\sum^{14}_{r=k}{r\choose k}{14\choose r}] = 6^7$$</span></p>
<p>my approach:</p>
<p>I tried to simplify the innermost sigma as well as trying to simplify by u... | epi163sqrt | 132,007 | <blockquote>
<p>Setting <span class="math-container">$n=7$</span> we obtain
<span class="math-container">\begin{align*}
\color{blue}{\sum_{k=0}^n}&\color{blue}{\binom{n}{k}\binom{2n}{k}^{-1}\sum_{r=k}^{2n}\binom{r}{k}\binom{2n}{r}}\\
&=\sum_{k=0}^n\binom{n}{k}\frac{k!(2n-k)!}{(2n)!}\sum_{r=k}^{2n}\frac{r!}{k!(r... |
2,705,980 | <p>I have the following problem:
\begin{cases}
y(x) =\left(\dfrac14\right)\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2 \\
y(0)=0
\end{cases}
Which can be written as:</p>
<p>$$ \pm 2\sqrt{y} = \frac{dy}{dx} $$</p>
<p>I then take the positive case and treat it as an autonomous, seperable ODE. I get $f(x)=x^2$ as my ... | Michael Burr | 86,421 | <p>Depending on how much you know, there are some quick proofs. If you know the invertible matrix theorem and basis theorem, then you can get this directly.</p>
<p>Since $\{A\bf{v}_1,\dots,A\bf{v}_n\}$ are linearly independent in $\mathbb{R}^n$, they form a basis. Therefore, the image of $A$ contains a basis for $\m... |
2,190,551 | <p>How can I find the degrees of freedom of a $n \times n$ real orthogonal matrix?</p>
<p>I have tried to proceed by principle of induction but I fail.Please tell me the right way to proceed.</p>
<p>Thank you in advance.</p>
| Community | -1 | <p>Answer to @Distracted Kerl and to the question asked by the OP.</p>
<p>There are essentially $2$ proofs of the required result; both uses the famous formula $1+2+\cdots+n=n(n+1)/2$. You chose the most difficult one. </p>
<p>$O(n)$ is an algebraic set. If $A\in O(n)$, then there is an orthogonal $P$ s.t. $P^TAP$ is... |
313,025 | <p>I got two problems asking for the proof of the limit: </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$</p>
</blockquote>
<p>and, </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,... | Julien | 38,053 | <p>I'll do the second one. And it appears that André Nicolas did the first one while I was writing, so everything is fine.</p>
<p>By integration by parts,
$$
x\int_0^{+\infty}\frac{e^{-px}}{p+1}dp=1-\int_0^{+\infty}\frac{e^{-px}}{(p+1)^2}dp\leq 1
$$
for all $x>0$.</p>
<p>Now by Lebesgue dominated convergence theo... |
3,245,428 | <p>Is it true that every tame knot has at least an alternating diagram?</p>
<p>If yes, is it true that we can always obtain an alternating diagram by a finite number of Reidemeister moves from a diagram of a knot? </p>
<p>If yes, how can we do it?</p>
<p>I am reading GTM Introduction to Knot Theory and find they sor... | Kyle Miller | 172,988 | <p>A knot is called <em>alternating</em> if it has an alternating knot diagram. If there is a sequence of Reidemeister moves on a diagram for a knot that results in an alternating diagram, then the knot is an alternating knot. Because Reidemeister moves are a complete set of moves, given a diagram for an alternating ... |
3,617,600 | <p>I am trying to understand the proof of the First and Second Variation of Arclength formulas for Riemannian Manifolds. I want some verifaction that the following covariant derivaties commute. I find it intuitive but I want to also have a formal proof.</p>
<p>Some notation: Let <span class="math-container">$\gamma(t,... | Nick A. | 412,202 | <p>HK Lee's answer is 100% correct. I am going to answer my question just to have an answer in the notation that I am more familiar with i.e. with connections along maps. </p>
<p>Say we have <span class="math-container">$f:(N,h)\rightarrow (M,g)$</span> and we endow <span class="math-container">$M$</span> with a conne... |
3,959,263 | <p>Let <span class="math-container">$G$</span> be a tree with a maximum degree of the vertices equal to <span class="math-container">$k$</span>.
<strong>At least</strong> how many vertices with a degree of <span class="math-container">$1$</span> can be in <span class="math-container">$G$</span> and why?</p>
<p>I think ... | Jonas Linssen | 598,157 | <p><strong>Hint</strong> Any tree on <span class="math-container">$\geq 2$</span> vertices has at least two leafs. Consider a tree with maximal degree <span class="math-container">$k$</span> and delete a vertex <span class="math-container">$v$</span> with degree <span class="math-container">$k$</span>. You are left wit... |
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Tony | 1,491 | <p>The assertion $(p+5)(p-1) \equiv 0 \pmod{16}$ is equivalent to $16 \mid (p+5)(p-1)$. Then you consider cases: $2^4 \mid (p+5)$, $2^3 \mid (p+5)$ and $2 \mid p-1$, $2^2 \mid p+5$ and $2^2 \mid p-1$, etc. </p>
|
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Mohamed | 33,307 | <p><strong>Hint :</strong> We can use the existence and unicity of decomposition of all non zero integer $N$ as $N=2^m q$ where $m$ is an integer and $q$ an odd integer.
We write :</p>
<p>$p+5=2^k u $ and $p-1 = 2^l v $ where $u$ and $v$ are odd, that implies $u2^k-5 = v 2^l +1$ implies $u2^k-v 2^l ... |
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | Jackson Walters | 13,181 | <p>$$(p+5)(p-1) \equiv 0 \text{ (mod 16)} $$
$$\Leftrightarrow (p+5)(p-1)=16k $$
$$\Leftrightarrow p^{2}+4p+(-5-16k)=0$$
$$\Rightarrow p=-2 \pm\frac{1}{2}\sqrt{36+64k}$$
$$\Rightarrow p=-2 \pm \sqrt{9+16k}$$</p>
<p>$$k=0: p=-2\pm 3 \Rightarrow p \in \{1,-5\} \equiv \{1,11\} \text{ (mod 16)}$$
$$k=1: p=-2 \pm 5 \Righta... |
1,981,948 | <p>Is there a relationship between group order and element order? </p>
<p>I know that there is a relationship between group order and subgroup order, which is that $[G:H] = \frac{|G|}{|H|}$ where $H$ is the subgroup of $G$ and $[G:H]$ is the index of $H$ in $G$. But is there a relationship between group order and th... | dezdichado | 152,744 | <p>The order of any element must divide the group order if the group is finite. This follows from what you have written i.e, the Lagrange's theorem. </p>
<p>Your second question is a consequence of the Cauchy's theorem.</p>
|
1,360,835 | <p>Reading "A First Look at Rigorous Probability Theory", and in the definition of outer measure of a set A, we take the infimum over the measure of covering sets for A from the semi-algebra (e.g., intervals in [0,1] ).</p>
<p>Is this set over which we are taking the infimum well-defined? For a given real
number x, ho... | pancini | 252,495 | <p>First of all, never assume what you are trying to prove. It is possible that it makes sense in your mind to assume something and then test to show it is true, but this is not how formal math is done so most instructors wouldn't accept it.</p>
<p>Second, I am a bit confused about what you are trying to show. This is... |
4,076,033 | <p>I know how to check if a vector or a matrix is linearly dependent or independent , but how do I apply it on this problem?</p>
<p>Let V1 , V2 , V3 be vectors
How do I prove that the vector V3 = ( 2, 5, -5) is linearly dependent on V1 = ( 1,-2,3) and V2 = (4,1,1) ?</p>
<p>Will it be enough or correct if I solved the e... | José Carlos Santos | 446,262 | <p>Yes, that will work.</p>
<p>Or you can check that<span class="math-container">$$\begin{vmatrix}2&1&4\\5&-2&1\\-5&3&1\end{vmatrix}=0.$$</span>It follows from this that there are numbers <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span> and <span class="m... |
341,602 | <p>Let <span class="math-container">$L$</span> be a semisimple Lie algebra and let <span class="math-container">$(V,\varphi)$</span> be a finite-dimensional <span class="math-container">$L$</span>-module representation. Our main goal is to prove that <span class="math-container">$\varphi$</span> is completely reducible... | F Zaldivar | 3,903 | <p>Since you also ask for another method, perhaps you may try Hans Samelson's approach in his textbook "Notes on Lie Algebras" (I have an <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=254112" rel="nofollow noreferrer">older edition</a> but it has been <a href="https://link.springer.com/book/10.1007/978-1-46... |
2,547,508 | <p>I am trying to prove that various expressions are real valued functions. Is it possible to state that, because no square roots (or variants such as quartic roots etc) are in that function, it is a real valued function?</p>
| B. Goddard | 362,009 | <p>Well, $\sin^{-1} 2 = 1.570796327-1.316957897i$ (for a usual branch.) I don't know if this counts as using square root. The series for $\sin^{-1}x$ is found by integrating the series for $1/\sqrt{1-x^2},$ so there's a square root floating around.</p>
|
2,172,836 | <p>I'm writing a small java program which calculates all possible knight's tours with the knight starting on a random field on a 5x5 board.</p>
<p>It works well, however, the program doesn't calculate any closed knight's tours which makes me wonder. Is there an error in the code, or are there simply no closed knight's... | Lelouch | 152,626 | <p>The definition of a 'closed' knights tour on a $m \times n$board, is a sequence of steps from a starting square $a_1$ to another square $a_{mn}$ , such that every square is visited exactly once, and the last sqaure is only one knight step away from $a_1$. Having said that, it is obvious, that for $mn $(mod2) $= 1$, ... |
4,042,741 | <p>I'm really struggling to understand the literal arithmetic being applied to find a complete residue system of modulo <span class="math-container">$n$</span>. Below is the definition my textbook provides along with an example.</p>
<blockquote>
<p>Let <span class="math-container">$k$</span> and <span class="math-conta... | fleablood | 280,126 | <p>Prelim: The definition is confusing because it is <em>not</em> assuming <span class="math-container">$k = n$</span>. You <em>will</em> be able to prove <span class="math-container">$k = n$</span> <em>later</em> but in mathematics we don't include <em>anything</em> in a definition that we can prove later.</p>
<p>The... |
1,403,486 | <p>As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that <span class="math-container">$\{(x,y): x>0\}$</span> is connected. My tools for this are:</p>
<blockquote>
<p><strong>Definition 1</strong>: Two dis... | graydad | 166,967 | <p>The two definitions really are saying the same thing. And not even in one of those weird ways where two things are equivalent but don't sound at all related. Like the Bolzano-Weierstrass Theorem.</p>
<p>For a proof by contradiction, let's suppose your definition of equal sets ($X \subseteq Y$ and $Y \subseteq X$) h... |
331,859 | <p>I need to find the antiderivative of
$$\int\sin^6x\cos^2x \mathrm{d}x.$$ I tried symbolizing $u$ as squared $\sin$ or $\cos$ but that doesn't work. Also I tried using the identity of $1-\cos^2 x = \sin^2 x$ and again if I symbolize $t = \sin^2 x$ I'm stuck with its derivative in the $dt$.</p>
<p>Can I be given a h... | lab bhattacharjee | 33,337 | <p>$$\text{ As }\cos2y=2\cos^2y-1=1-2\sin^2y$$</p>
<p>$$\sin^6x\cos^2x=\left(\frac{1-\cos2x}2\right)^3\left(\frac{1+\cos2x}2\right)$$</p>
<p>$$16\sin^6x\cos^2x=(1-3\cos2x+3\cos^2x-\cos^32x)(1+\cos2x)$$</p>
<p>$$=\left(1-3\cos2x+3\frac{(1+\cos4x)}2-\frac{(\cos6x+3\cos2x)}4\right)(1+\cos2x)$$ (applying $\cos3y=4\cos^... |
366,724 | <p>Suppose $D$ is an integral domain and that $\phi$ is a nonconstant function from $D$ to
the nonnegative integers such that $\phi(xy) = \phi(x)\phi(y)$. If $x$ is a unit in $D$, show that
$\phi(x) = 1$.</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> First show that if $e$ is the identity element, then $\phi(e)=1$. This should be an easy consequence of $ee=e$. </p>
<p>Then use the fact that if $x$ is a unit, and $y$ is the inverse of $x$, then $\phi(e)=\phi(xy)=\phi(x)\phi(y)$. </p>
<p><strong>Added:</strong> It is all too easy to forget... |
696,869 | <p>Question:
Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible.</p>
<p>My work: Based on the section I read, I will treat I to be an identity matrix, which is a $1 \times 1$ matrix with a $1$ or as an square matrix with main diagonal is all ones and the rest is zer... | vadim123 | 73,324 | <p>The inverse of $A$ is almost certainly not $A$. You have $A^2+2A=-I$, or $A(A+2I)=-I$. Multiplying by $-1$ you get $$A(-A-2I)=I$$</p>
<p>Hence the inverse of $A$ is $-A-2I$.</p>
|
696,869 | <p>Question:
Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible.</p>
<p>My work: Based on the section I read, I will treat I to be an identity matrix, which is a $1 \times 1$ matrix with a $1$ or as an square matrix with main diagonal is all ones and the rest is zer... | user76568 | 74,917 | <p>$$\det{A} \cdot \det{(A+2I)} =\det{[A(A+2I)]}=\det{(-I)}=\pm 1 \implies \det{A} \neq 0 \iff A \space \text{is invertible}$$ </p>
|
2,898,767 | <p>For $M_n (\mathbb{C})$, the vector space of all $n \times n $ complex matrices,</p>
<p>if $\langle A, X \rangle \ge 0$ for all $X \ge 0$ in $M_n{\mathbb{C}}$,then $A \ge 0$</p>
<p>which of the following define an inner product on $M_n(\mathbb{C})$?</p>
<p>$1)$$ \langle A, B\rangle = tr(A^*B)$</p>
<p>$2)$$ \... | user | 505,767 | <p>The fact is that <a href="http://www.wolframalpha.com/input/?i=domain%20x%5E(1%2F3)" rel="nofollow noreferrer"><strong>Wolfram</strong></a> assumes the domain $x\ge 0$ for the function $\sqrt[3] x$ even if for $x \in \mathbb{R}$ the function is well defined on the whole domain.</p>
|
402,750 | <p>I read that a continuous random variable having an exponential distribution can be used to model inter-event arrival times in a Poisson process. Examples included the times when asteroids hit the earth, when earthquakes happen, when calls are received at a call center, etc. In all these examples, the expected value ... | Tim | 74,128 | <p>The link between the Poisson point process and the exponential is probably the key to understanding this.</p>
<p>Let's say that asteroids hit your planet as a Poisson point process rate $\lambda$ per minute. This means that if $X_i$ is the number of asteroids that arrive in the $i$th minute then the sequence $X_i$... |
402,750 | <p>I read that a continuous random variable having an exponential distribution can be used to model inter-event arrival times in a Poisson process. Examples included the times when asteroids hit the earth, when earthquakes happen, when calls are received at a call center, etc. In all these examples, the expected value ... | not all wrong | 37,268 | <p>The key property of the exponential distribution is that it is memoryless, so that translation of the PDF in time $x\to x-c$ is just scaling of the PDF $f\to e^{\lambda c} f$ plus cutting it off at zero. Probabilistically this says $P(X>s+t|X>s)=P(X>t)$. That is, it doesn't matter how long it was since the ... |
402,750 | <p>I read that a continuous random variable having an exponential distribution can be used to model inter-event arrival times in a Poisson process. Examples included the times when asteroids hit the earth, when earthquakes happen, when calls are received at a call center, etc. In all these examples, the expected value ... | Raskolnikov | 3,567 | <p>Just restating my comment as an answer:</p>
<p>You are confusing "long interarrival times are rare" with "the more time passes by, the more likely it is an earthquake will happen". The first is true for an exponential distribution if your mean interarrival time is short. The second one is never true for an exponent... |
296,536 | <p>Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)</p>
<p>Moreover generally I would like to ... | Willie Wong | 3,948 | <p>The limit doesn't always exist. </p>
<p>For convenience I will use the Lebesgue measure; similar construction can also be done for most $\mu$. </p>
<p>Let $c_n$ be a sequence of increasing positive integers. Consider the function
$$ f(x) = \sum_{n = 0}^\infty c_n^2 \chi_{[n, n + (c_n!)^{-1}]}(x) $$</p>
<p>By th... |
1,032,926 | <p>I'm trying to understand this proof of the following Lemma, that I found in an article on Existence of Eigenvalues and Eigenvectors, but I don't understand the following step:</p>
<p><em>Let $V$ be a finite-dimensional complex vector space, $v\in V$ and $c\gt 0$. Since for every $v\in V\setminus \{ 0 \}$ and $k\in\... | Robert Lewis | 67,071 | <p>We need to assume $c > 0$. So assuming, we then have $\Vert (T - k I)(v) \Vert \ge c\Vert v \Vert > 0$ for $v \ne 0$. Thus $(T - k I)v \ne 0$ for $v \ne 0$. If now $T - kI$ were <em>not</em> injective, then there would exist <em>some</em> $w \in V$ and <em>distinct</em> $y_1, y _2 \in V$ with $(T - k I)y_1 ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.