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 |
|---|---|---|---|---|
560,234 | <p>Given the series</p>
<blockquote>
<p>$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$</p>
</blockquote>
<p>I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$ might not exist, so the series is ... | Mr.Fry | 68,477 | <p>Apply $n^{th}$ test, which states, if $\lim_{\ n\to\infty} a_n\neq0$ then $\sum_{n=0}^{\infty}a_n$ diverges.</p>
|
241,586 | <p>It is perhaps well known that the sign function is discontinuous, if defined for $f:\mathbb{R}\rightarrow \mathbb{R}$. However, if we were to define the sign function for $f:\mathbb{R} \setminus \left \{ 0 \right \}\rightarrow \mathbb{R}$, would the sign function still remain discontinuous? </p>
<p>My belief is yes... | Hagen von Eitzen | 39,174 | <p>If we expell $0$ from the domain, the sign function becomes continuous (in fact, it is even <em>locally constant</em>). For $x\ne 0$ (and $\epsilon>0$) pick $\delta=|x|$. Then you have $f(y)=f(x)$ for all $y$ with $|y-x|<\delta$.</p>
<p>More trivia: Contrary to popular belief, the function given by $f(x)=\fra... |
632,029 | <blockquote>
<p>Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous such that
$$\lim_{x\to\infty} f(x) = \lim_{x\to -\infty}f(x) = 0$$
Show that there $\exists x_0\in\mathbb{R}: \, \mid f(x)\mid \, ≤ \, \mid f(x_0)\mid$ for $\forall x \in \mathbb{R}$.</p>
</blockquote>
<p>Basically, this means I have to sho... | Igor Rivin | 109,865 | <p>The conditions mean that for every $\epsilon > 0,$ there exists an $R>0,$ such that $|f(x)| < \epsilon$ for $|x|>R.$ On the compact set $[-R, R],$ you know that $|f|$ has a maximum $M.$ If $M > \epsilon,$ that is the global maximum. If not, repeat the argument with $M/2$ in place of $\epsilon.$ This ... |
1,877,632 | <p>My integral calculus is rusty.
How do I calculate the interior area (blue region) of four bounding circles?<br><br>
<a href="https://i.stack.imgur.com/VtQIy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VtQIy.png" alt="enter image description here"></a></p>
| gerw | 58,577 | <p>Hint: Draw a square around your circles. Then you do not need any integrals, only the area of the circles (and of the square).</p>
|
2,487,297 | <p>I am reading Tu's book "Introduction to Manifolds". In Example 11.8, it says that $f(t) = (t^2 - 1,t^3-t)$ is an immersion however it says that the equation $f'(t) = (2t,3t^2 -1) = (0,0)$ has no solution in $t$ and this means the map $f$ is an immersion. This confuses me since I thought the definition of an immersio... | A. Goodier | 466,850 | <p>$f:\mathbb{R}^n\to\mathbb{R}^m$ is an immersion if the rank of the matrix of the linear map $d_xf:T_x\mathbb{R}^n\to T_{f(x)}\mathbb{R}^m$ is $n$ for every $x\in\mathbb{R}^n$. This is equivalent to $d_xf$ being injective for every $x\in\mathbb{R}^n$.</p>
<p>My notation for the differential (your $f'(t)$) is $d_tf$.... |
3,193,823 | <p>I am asked to evaluate: <span class="math-container">$\frac{4+i}{i}+\frac{3-4i}{1-i}$</span></p>
<p>The provided solution is: <span class="math-container">$\frac{9}{2}-\frac{9}{2}i$</span></p>
<p>I arrived at a divide by zero error which must be incorrect. My working:</p>
<p><span class="math-container">$\frac{4+... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: It is <span class="math-container">$$\frac{4+i}{i}+\frac{3-4i}{1-i}=\frac{(4+i)(1-i)+(3-4i)i}{i(1-i)}$$</span></p>
|
3,193,823 | <p>I am asked to evaluate: <span class="math-container">$\frac{4+i}{i}+\frac{3-4i}{1-i}$</span></p>
<p>The provided solution is: <span class="math-container">$\frac{9}{2}-\frac{9}{2}i$</span></p>
<p>I arrived at a divide by zero error which must be incorrect. My working:</p>
<p><span class="math-container">$\frac{4+... | Aaratrick | 555,919 | <p>You have made a mistake in the second last step of your simplification.
<span class="math-container">$$1 - i^2 = 1 - (-1)=1 + 1 = 2$$</span>
Then,
<span class="math-container">$$-4i + 1 + \frac{7 - i}{2}
= 1 + \frac{7}{2} + i(-4 - \frac{1}{2})
= \frac{9}{2} - \frac{9}{2}i$$</span></p>
|
201,876 | <p>What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? Illustration from Wikipedia <img src="https://i.stack.imgur.com/oWM54.png" alt="enter image description here"> </p>
<p... | Leandro Vendramin | 17,845 | <p>One very nice family of examples is the one of <strong>set-theoretical solutions</strong>. </p>
<p>In the paper</p>
<ul>
<li>Drinfelʹd, V. G. On some unsolved problems in quantum group theory. Quantum groups (Leningrad, 1990), 1--8, Lecture Notes in Math., 1510, Springer, Berlin, 1992. MR1183474 (94a:17006)</li>
<... |
410,763 | <p>I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any formula? As far as I can recall, maybe I can write</p>
<p><span class="math-container">$$\int \nabla \times \vec{A} \ \... | Shuhao Cao | 7,200 | <p>Yes, it is right except the the sign on the right hand side should be minus. It can be viewed as a Corollary to the Gauss-Green Theorem (Evans's PDE book appendix C.2 uses this name):
$$
\int_{\Omega} \partial_{x_i} u\, dx = \int_{\partial \Omega} u n_i \,d\sigma.
$$
The formula you gave is essentially:
$$
\int_{\Om... |
2,411,890 | <blockquote>
<p>If $x$ is a real number, then $|x+1| \leq 3$ implies that $-4 \leq x \leq 2$.</p>
</blockquote>
<p>I've tried to prove this by exhaustion, is that the right way to prove it? </p>
| Fred | 380,717 | <p>If $a$ and $b$ are real numbers and if $b \ge 0$, then the inequality $|a| \le b$ exactly means</p>
<p>$$-b \le a \le b.$$</p>
<p>Hence , if $b=3$, then</p>
<p>$$|a| \le 3 \iff -3 \le a \le 3.$$</p>
<p>Now let $a=x+1$. Then we get</p>
<p>$$|x+1| \le 3 \iff -3 \le x+1 \le 3 \iff -4 \le x \le 2.$$</p>
|
3,561,805 | <p>Problem: Let <span class="math-container">$W$</span> equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose <span class="math-container">$P(W<1)=0.02$</span> and <span class="math-container">$P(W>1.072)=0.08$</span>. Call a box of soap light, good, or heavy dependin... | Robert Lewis | 67,071 | <p>With <span class="math-container">$n$</span> odd, we have</p>
<p><span class="math-container">$n = 2k + 1. \; k \in \Bbb Z; \tag 1$</span></p>
<p>then</p>
<p><span class="math-container">$n^2 = 4k^2 + 4k + 1, \tag 2$</span></p>
<p>whence</p>
<p><span class="math-container">$n^2 - 1 = 4k^2 + 4k = 4(k^2 + k) \Lon... |
4,274,526 | <p>When proving a limit at <span class="math-container">$a$</span> with value <span class="math-container">$L$</span> with the definition, we must show that for all <span class="math-container">$\epsilon >0$</span>, there is <span class="math-container">$\delta >0$</span> such that:</p>
<p><span class="math-conta... | José Carlos Santos | 446,262 | <p>No. Suppose that <span class="math-container">$f=\cos$</span> (with domain equal to <span class="math-container">$(-1,1)$</span>), that <span class="math-container">$L=2$</span> and that <span class="math-container">$a=0$</span>. Then we always have <span class="math-container">$|x-a|<\bigl|f(x)-L\bigr|$</span>, ... |
2,433,438 | <p>Probability </p>
<blockquote>
<p>Question. "$11$ identical balls are distributed in $4$ distinct boxes randomly. Then the probability that any $3$ boxes will together get a greater number of balls than the remaining one is:"</p>
</blockquote>
<p>I am confused about the distribution of balls. Are all the cases i.... | ploosu2 | 111,594 | <p>The solution given by OnoL has the correct configurations for the ball amounts in the boxes but it is missing the possible ways the balls can be placed to the boxes. Because we are considering all the ways $11$ balls can be placed to $4$ boxes ($4^{11}$) these multipliers must be considered. They are also different ... |
316,672 | <p>I do not see how this is even valid. Could someone point this out to me:</p>
<p>Assume that $x_n$ is a cauchy sequence of rational numbers satisfying $|x_n| \geq r$ for all $n\in\mathbb{N}$. Show that there is $N\in\mathbb{N}$ s.t. either $x_n > r$ for all $n \geq N$ or $x_n < -r$ for all $n\geq N$.</p>
<p>H... | gnometorule | 21,386 | <p>Your objection is correct if you qualify "constant at $r$" (if it is constant at any other value, it holds vacuously). However, write this as part 1 of your answer; then show in part 2 that it is true if $x_n \neq r \,$ eventually. </p>
|
419,205 | <p>My question is: is it possible to find pentagonal numbers which are also tetrahedral?
A pentagonal number is obtained by the formula:
$$P_k=\frac{1}{2}k(3k-1)$$
The equivalent formula for the tetrahedral number $T_n$
is:
$$T_n=\frac{1}{6}n(n+1)(n+2)$$
So the problem is to find a $T_n=P_k$ that means to solve:
$$n(n+... | egreg | 62,967 | <p>An eigenvalue of the $n\times n$ matrix $A$ is a number $\lambda$ such that there is a nonzero vector $v$ with
$$
Av=\lambda v.
$$
Depending on the context you may be looking for $\lambda$ in the real or complex numbers. In any case, the equation is equivalent to
$$
(A-\lambda I)v=0
$$
where $I$ is the $n\times n$ i... |
4,298,602 | <p><a href="https://i.stack.imgur.com/jX5H1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jX5H1.png" alt="enter image description here" /></a></p>
<p>The prime <strong>Cauchy</strong> product is <span class="math-container">$C_n = (a_1b_1) + (a_1b_2+a_2b_1) + (a_1b_3+a_2b_2+a_3b_1)+...$</span>，
In ... | Ian | 83,396 | <p>Maybe a different notational framework presenting the same ideas will help you understand the proof.</p>
<p>We begin by introducing introducing <span class="math-container">$\varepsilon_1,\varepsilon_2,\varepsilon_3$</span> and associated <span class="math-container">$N_1,N_2,N_3$</span> based on the facts that <spa... |
2,496,309 | <p>i am wondering if the unity elements of a ring form a ring ? In other words do they form an abelian group under addition ? I have tryied but i have not reached to a conclusive answer. Thanks for any comment.</p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>The limit is the inverse of</p>
<p>$$\lim_{h \to 0}\frac{\sqrt{4+h}-\sqrt4}{h},$$ which should ring a bell.</p>
|
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| rschwieb | 29,335 | <p>The main thing to notice is that it is not always, as a student might first guess, just $\{ab\mid a\in I, b\in J\}$. That works for groups, but in a ring you have two operations going on. Certainly in addition to having all the pairwise products, it would also have to have all possible sums of those products. Otherw... |
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| Community | -1 | <p>For a complete answer let me add an example: $I=(2,X)$ and $J=(3,X)$ in $\mathbb Z[X]$. Then $IJ=(6,X)$ (why?), thus $X\in IJ$ and $X$ can't be written as $ij$ with $i\in I, j\in J$ (why?). (Note that if one of the ideals is principal one can't get such an example.)</p>
|
627,749 | <blockquote>
<p>Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial.</p>
</blockquote>
<p>OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})</p>
| Saar peer | 119,190 | <p>Let $g \in G$ since $G'=G$ then exist $g_1, g_2 \in G$ such that $g_1g_2g_1^{-1}g_2{^-1} = g$.</p>
<p>Then we get : $f(g) = f(g_1g_2g_1^{-1}g_2^{-1}) = f(g_1)f(g_2)f(g_1^{-1})f(g_2^{-1}) ∈ A' = \{e\}$</p>
|
3,064,458 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow\mathbb{R}$</span> be a continuous function. </p>
<ol>
<li>Show that for each <span class="math-container">$\epsilon\in(0,1)$</span>, <span class="math-container">$\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon... | Aphelli | 556,825 | <p>Okay, so please let me state an analysis principle that will be very useful in numerous problems.</p>
<p>Interversion is painful. </p>
<p>Either there is a straightforward (« logically tautological », that is, that does not require any analysis) argument for doing it, or it requires thinking the whole reasoning th... |
3,064,458 | <blockquote>
<p>Let <span class="math-container">$f:[0,1]\rightarrow\mathbb{R}$</span> be a continuous function. </p>
<ol>
<li>Show that for each <span class="math-container">$\epsilon\in(0,1)$</span>, <span class="math-container">$\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon... | Ran Kiri | 632,309 | <p>So you managed to show that for each <span class="math-container">$\varepsilon\in(0,1)$</span> you get:
<span class="math-container">$$ \lim\limits_{n\to\infty}\int\limits_0^{1-\varepsilon}f(x^n)\,{\rm d}x=(1-\varepsilon)f(0) $$</span>
So for each <span class="math-container">$\varepsilon\in(0,1)$</span> you can now... |
724,462 | <p>I can not solve this question
Find a compound proposition logically equivalent to $p \to q$ using only the logical operator $\downarrow$.</p>
| Saaqib Mahmood | 59,734 | <p>Given propositions <span class="math-container">$p$</span> and <span class="math-container">$q$</span>, the propositon <span class="math-container">$p \downarrow q$</span> is given by the following truth table:</p>
<p><span class="math-container">$$
\begin{align}
p \qquad & q & p \downarrow q \\
T \qquad &... |
3,916,490 | <p>It's easy to get this:
<span class="math-container">$$\int \sqrt{1+\sin x}\, dx \\= \int \sqrt{ \sin^2{\frac{x}{2}} + \cos^2{\frac{x}{2}} + 2\sin{\frac{x}{2}}\cos{\frac{x}{2}}}\,\, dx \\ = \int \left | \sin{\frac{x}{2}} + \cos{\frac{x}{2}} \right |\, dx \\= \sqrt{2} \int \left | \sin{\left ( \frac{x}{2} + \frac{\pi... | zkutch | 775,801 | <p>Hint: <span class="math-container">$|\sin x|=(-1)^n \sin x$</span> for <span class="math-container">$\pi n \leqslant x < \pi (n+1)$</span>, where <span class="math-container">$n=0, \pm 1,\pm2, \cdots$</span>.</p>
<p>To keep antiderivative continuous in <span class="math-container">$x=\pi (n+1)$</span> you need to... |
1,493,874 | <p>How can solve that logarithms</p>
<p>$\log _{\frac{4}{x}}\left(x^2-6\right)=2$</p>
<p>It's look diffucult to solve </p>
<p>I was solve but stop with</p>
<p>$x^4−6x^2−16=0$</p>
<p>what is next?</p>
| Alice Ryhl | 132,791 | <p>Hint: raise $4/x$ to both sides</p>
<p>\begin{align}
\log_{\frac{4}{x}}\left(x^2-6\right)&=2\\
\left(\frac4x\right)^{\left(\log_{\frac{4}{x}}\left(x^2-6\right)\right)}&=\left(\frac4x\right)^2\\
\end{align}</p>
|
1,089,593 | <p>How to solve $\dfrac{dy}{dx}=\cos(x-y)$ ? How do I separate x and y here ?</p>
<p>Please advise.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>with $$u=x-y$$ we get $$y'=1-u'$$ nand our equation will be
$$1-u'=\cos(u)$$</p>
|
31,701 | <p>How do I write a chevron/circum (^) in MathJax? A backslash doesn't work as an escape character.</p>
<p>(Specific context: using <em>x</em>^<em>y</em> to mean XOR as in <span class="math-container">$x\oplus y$</span>. It's not my choice of notation so don't tell me that I can just use a different symbol.)</p>
| Jyrki Lahtonen | 11,619 | <p>Is <span class="math-container">$x\hat{}y$</span> acceptable? Gotten with <code>$x\hat{}y$</code>. In-line I have used
<code>$x$^$y$</code> which produces <span class="math-container">$x$</span>^<span class="math-container">$y$</span>. To get that effect in a displayed formula, an obvious work around is to wrap it ... |
255,652 | <p>I came across a problem that says:</p>
<p>Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?</p>
<blockquote>
<p>(a) The function $f$ is bounded.<br>
(b) The limit $\lim_{x\to\infty}f(x)$ exists.<br>
(c) The function $f$ is unifor... | Espen Nielsen | 45,874 | <p>You can at least eliminate a), b) and d).
Let $f_1(x)=x$. Then $f'(x)=1$ for all $x$ and $|f'|$ is therefore bounded. This eliminates a) and b). Now let $f_2(x)=0$. Then $f'(x)=0$ for all $x$ and $|f'|$ is bounded. This eliminates d).</p>
|
255,652 | <p>I came across a problem that says:</p>
<p>Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?</p>
<blockquote>
<p>(a) The function $f$ is bounded.<br>
(b) The limit $\lim_{x\to\infty}f(x)$ exists.<br>
(c) The function $f$ is unifor... | checkmath | 25,077 | <p>Consider the function $$f(x)=x\cos(\ln (x^2+1)^{1/2}).$$</p>
<p>And verify (exercise) which conditions it satisfies. </p>
|
502,160 | <p>Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.</p>
<p>$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$</p>
<p>and by the sentence in brackets I mean that the $a_n$'s are not just mostly equal to $1$... | pathfinder | 23,431 | <p>Not sure if this satisfies your assumptions, but this is an interesting infinite product for <span class="math-container">$|z|<1$</span>,
<span class="math-container">$$e^z=\prod_{k=1}^\infty (1-z^k)^{-\frac{\mu(k)}{k}},$$</span>
where <span class="math-container">$\mu(k)$</span> is the Möbius function. <a href="... |
1,584,653 | <p>Let linear transformation is defined as </p>
<p>$\mathcal{A}(1,1,1)=(1,0,0)$</p>
<p>$\mathcal{A}(1,-1,0)=(1,1,1)$</p>
<p>$\mathcal{A}(1,0,1)=(1,1,1)$</p>
<p>Find matrix of $\mathcal{A}$ and inverse (not in matrix representation, if exists).</p>
<p>Attempt:</p>
<p>Transformation $\mathcal{A}$ can be represented... | Emilio Novati | 187,568 | <p>Let
$$
A=\begin{bmatrix}
a_1&b_1&c_1\\
a_2&b_2&c_2\\
a_3&b_3&c_3\\
\end{bmatrix}
$$</p>
<p>we have:
$$
A \begin{bmatrix}1\\1\\1 \end{bmatrix}=\begin{bmatrix}1\\0\\0 \end{bmatrix}
\Rightarrow \begin {cases}
a_1+b_1+c_1=1\\
a_2+b_2+c_2=0\\
a_3+b_3+c_3=0\\
\end {cases}
$$ </p>
<p>$$
A \begin{b... |
1,639,390 | <p>I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know <em>why</em> it's that exactly.</p>
<p>Also, I am having troubles justifying htat $\frac{n!}{k!} = n(n-1)(n-2)....k(k+1)$</p>
| Bernard | 202,857 | <p>Start from
$$(x^n)^{(k)}=\dfrac{n!}{(n-k)!}x^{n-k},\quad\text{which is easy to prove by induction.}$$
and apply it replacing $k$ with $n-k$:
$$(x^n)^{(n-k)}=\dfrac{n!}{(n-(n-k))!}\,x^{n-(n-k)}=\dfrac{n!}{k!}\,x^{k}. $$</p>
<p>Also
$$\frac{n!}{k!}=\frac{n(n-1)\cdots(k+1)k(k-1)\cdots1}{k(k-1)\cdots1}=n(n-1)\cdots(k+1... |
1,487,878 | <blockquote>
<p>Prove: $\sqrt[4]{4}$ is irrational</p>
</blockquote>
<p>I know that $\sqrt{p}$ is irrational where $p$ is a prime number.</p>
<p>So $\sqrt[4]{4}=\sqrt[4]{2*2}=16*\sqrt[4]{2}=16*2^{\frac{1}{2}^\frac{1}{2}}$ </p>
<p>What can I say about $2^{\frac{1}{2}^\frac{1}{2}}$?</p>
| Zelos Malum | 197,853 | <p>That is wrong, it is
$$\sqrt[4]{4}=\sqrt[4]{2^2}=\sqrt{2}$$
so it is irrational.</p>
|
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | R Davies | 394,895 | <p>I applaud your mathematical enthusiasm! My preference once I started my maths degree was for applied things, so the maths behind the weather! Try that as a research starting point and see where it takes you.</p>
<p>Else I second the recommendation for programming, I got taught c++ on my maths degree, and getting co... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Colonel Panic | 6,450 | <p>Solving problems is fun. </p>
<blockquote>
<p>It used to be said that mathematics and cricket were not spectator sports; and this is still true of mathematics.
To progress as a mathematician, you have to strengthen your mathematical muscles. It is not
enough just to read books or attend lectures. You have to ... |
4,624,011 | <p>I am looking to minimize the value of:
<span class="math-container">$$g(t)=\mathrm{Tr}\left[\exp(X+tY)\right]$$</span>
where both <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are symmetrical matrices with real coefficients. In general, <span class="math-container">$X$</span> an... | Robert Israel | 8,508 | <p>We have <span class="math-container">$$\exp(X+tY) = \sum_{n=0}^\infty (X+tY)^n/n! $$</span>
Now each <span class="math-container">$(X+tY)^n$</span> can be expanded as a sum of products of <span class="math-container">$n$</span> terms, where each term is either <span class="math-container">$X$</span> or <span class="... |
153,772 | <p>I am reading a paper on Chiral Differential Operators</p>
<p><a href="http://arxiv.org/pdf/hep-th/0604179v3.pdf" rel="nofollow noreferrer">http://arxiv.org/pdf/hep-th/0604179v3.pdf</a></p>
<p>and it says on page 23 that a line bundle over a manifold C can be characterized completely by its restriction to a non-tri... | Alex Degtyarev | 44,953 | <p>The statement is not true: the bundle is characterized by its restriction to <strong>all</strong> $2$-cycles, i.e., by its $2$-cohomology class (Chern class $c_1$). This is a simple fact in algebraic topology. One (the easiest, but not the most elementary) way to prove it is the exponential exact sequence $0\to\math... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Mr Pie | 477,343 | <p>How about <span class="math-container">$$\pi \simeq \sqrt [3]{\cfrac{31}{1-\cfrac{12}{39^3-40}}}\tag{11 d.p.}$$</span> Easy to remember because <span class="math-container">$\pi^3\approx 31$</span> and <span class="math-container">$39$</span> is the number before <span class="math-container">$40$</span>.</p>
<p>Als... |
1,407,183 | <p>I know this is trivial but I don't don't know if I'm right</p>
<p>Verify, by substitution, if the function is a solution of differential equation.</p>
<p>$y'=3x^2$ , $y=x^3+7$</p>
<hr>
<p>Differentiating the function $y=x^3+7$ you get $y'=3x^2$</p>
<p>Adding that resut to the $y'=3x^2$ I get</p>
<p>$3x^2+3x^2... | jameselmore | 86,570 | <p>You have $y' = 3x^2$, and want to verify that $y_p = x^3 + 7$ is a solution to the equation.</p>
<p>Clearly, $y'_p = 3x^2$, so $y'_p = 3x^2 = y'$... it is a solution to the equation. In fact, $y_p = x^3 + C$ is a solution for all $C$ for the same reasoning above.</p>
|
2,522,255 | <p>I'm having difficulties to understand how to approach to this question:</p>
<p>Rolling a dice for 2 times.</p>
<ul>
<li>X - Result of the first time.</li>
<li>Y - The highest results from both of the rolls. </li>
</ul>
<p>Example: if in the first we'll have 5, and in the second 2, Y will be 5.</p>
<ol>
<li>Show ... | Graham Kemp | 135,106 | <p>Well, clearly $\mathsf P(X=k)~=~\begin{cases}1/6 &:& k\in\Bbb N, 1\leq k\leq 6\\0&:&\textsf{otherwise}\end{cases}$</p>
<p>And $\mathsf P(Y=j\mid X=k) ~=~ \begin{cases}\underline{\phantom{k/6}} &:& j\in\Bbb N,k\in\Bbb N, 1\leq k=j\leq 6 \\ \underline{\phantom{1/6}} &:& j\in\Bbb N,k\in... |
2,433,051 | <p>Following <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/29/" rel="nofollow noreferrer">Wolfram Sine inequalities</a> I found that</p>
<p>$$|\sin(x)| \le |x| \quad \text{for} \quad x \in \mathbb{R}$$
How can I prove this relation?</p>
| szw1710 | 130,298 | <p>For $x>0$ define $g(x)=\sin x-x$ and show that $g$ decreases. For $x<0$ use oddiness of $g$. For $x=0$ it is trivial.</p>
|
2,433,051 | <p>Following <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/29/" rel="nofollow noreferrer">Wolfram Sine inequalities</a> I found that</p>
<p>$$|\sin(x)| \le |x| \quad \text{for} \quad x \in \mathbb{R}$$
How can I prove this relation?</p>
| José Carlos Santos | 446,262 | <p>If $x\in\mathbb R$, $\bigl|\sin'(x)\bigr|=\bigl|\cos(x)\bigr|\leqslant1$. Therefore, by the mean value theorem,$$\bigl|\sin(x)\bigr|=\bigl|\sin(x)-\sin(0)\bigr|\leqslant1\times|x-0|=|x|.$$</p>
|
2,433,051 | <p>Following <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/29/" rel="nofollow noreferrer">Wolfram Sine inequalities</a> I found that</p>
<p>$$|\sin(x)| \le |x| \quad \text{for} \quad x \in \mathbb{R}$$
How can I prove this relation?</p>
| Simply Beautiful Art | 272,831 | <p>Note that the sine of theta is less than the arc-length. Pretty standard approach to evaluating the limit:</p>
<p>$$\lim_{x\to0}\frac{\sin(x)}x$$</p>
<p><a href="https://i.stack.imgur.com/7kMXN.png" rel="noreferrer"><img src="https://i.stack.imgur.com/7kMXN.png" alt="enter image description here"></a></p>
<p>This... |
441,994 | <p>In thinking about my question <a href="https://mathoverflow.net/questions/441982">here</a> for the Linial arrangement, the following limit arose:
<span class="math-container">$$ \lim_{n\to\infty}\frac{(n-1)\sum_{k=0}^n {n\choose k}(k+1)^{n-2}}
{\sum_{k=0}^n {n\choose k}(k+1)^{n-1}}. $$</span>
Is this limit ... | Nemo | 82,588 | <p>First of all, it seems like the value of the limit is more like <span class="math-container">$1.27...$</span>, and not <span class="math-container">$2.27...$</span>.</p>
<p>Using some heuristics outlined below it is possible to find the limit:
<span class="math-container">$$
a=1.278464542761...,
$$</span>
where <sp... |
1,336,175 | <p>I would like to study <strong>category of sets and multi-valued functions</strong>: A category whose objects are sets and morphisms are multi-valued functions. </p>
<p>By a multi-valued function $f:A\rightarrow B$, from set A to set B, I mean a function that assigns to each element of A, a subset of B. There might ... | Arno | 128,989 | <p>There are two categories you might be thinking of.</p>
<p>1) The category Rel of sets and relations, with composition defined via:
$z \in (P \circ Q)(x) \Leftrightarrow \exists y \in Q(x) \ z \in P(y)$</p>
<p>This category is described in Giorgio Mossa's answer.</p>
<p>2) The category Mult of sets and multivalued... |
2,168,524 | <p>I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.</p>
<p>But I can't figure out what is the reasoning behind this. Any help would be really appreciated! </p>
| MJD | 25,554 | <p>A multiple of 15 and a multiple of 21 are both multiples of 3.</p>
<p>The difference between two multiples of 3 is another multiple of 3.</p>
<p>1 is not a multiple of 3.</p>
|
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Julián Aguirre | 1,168 | <p>Ordinary Differential Equations by Philip Hartman, John Wiley & sons, 1964 might be what you are looking for.</p>
|
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | Kevin Pond | 30,557 | <p>Here's a nice book by Gerald Teschl.</p>
<p><a href="http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf" rel="nofollow">http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf</a></p>
|
81,221 | <p>Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations c... | guest | 109,816 | <p>My advice is different. Work through a standard undergrad text on DiffyQs first. Not one with all the fancy connections to other fields of math that you know. But one emphasizing manipulation and problem solving and applications. Then after that, go grab some fancy book with all the grad school emphasis on proofs... |
3,795,174 | <p>Let <span class="math-container">$a_n$</span> be a positive sequence.</p>
<p>We define <span class="math-container">$b_n$</span> as following:</p>
<p><span class="math-container">$$b_n = \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$</span></p>
<p><strong>Question:</strong> Prov... | Community | -1 | <p>We can write</p>
<p><span class="math-container">$$b_n=c_1+c_2+c_3+\cdots c_{n-1}+\frac1{c_1c_2c_3\cdots c_{n-1}}$$</span> where the <span class="math-container">$c_k$</span> are positive numbers.</p>
<p>The minimum value of <span class="math-container">$b_n$</span> is found by cancelling the gradient,</p>
<p><span ... |
3,096,741 | <p><span class="math-container">$X, Y$</span> are two independent <span class="math-container">$\mathcal{N}(0,1)$</span> random variables </p>
<p>this question was a follow up question of this <a href="https://math.stackexchange.com/questions/3096530/show-that-e-fracx22-in-l1-iff-exy-in-l1-iff-exy-in-l1">one</a></p>
... | Did | 6,179 | <p>Since <span class="math-container">$Y$</span> is standard normal, <span class="math-container">$E(e^{xY})=e^{x^2/2}$</span> for every <span class="math-container">$x$</span>, hence, by independence of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, <span class="math-container">$E... |
2,610,210 | <blockquote>
<p>solve using DE with integrable solution
$$
y(y^2-2x)dx + x(y^2+x)dy = 0
$$</p>
</blockquote>
| Claude Leibovici | 82,404 | <p><strong>Welcome to the site !</strong></p>
<p>You must understand that a lot of people are ready to help you <strong>provided</strong> that you explain (and show) what you already tried and tell where you are stuck.</p>
<p>As a welcome gift, I shall put you on the track.</p>
<p>Consider the differential equation ... |
2,610,210 | <blockquote>
<p>solve using DE with integrable solution
$$
y(y^2-2x)dx + x(y^2+x)dy = 0
$$</p>
</blockquote>
| JJacquelin | 108,514 | <p>HINT : </p>
<p>The change of variables $\quad xy=u\quad \text{and}\quad\frac{x^2}{y}=v \quad $ leads to $\quad du-dv=0\quad\to\quad u-v=c$
$$xy-\frac{x^2}{y}=c \quad\to\quad y=\frac{c\pm\sqrt{c^2+4x^3}}{2x}$$</p>
<p>More classical way : the integrating factor method. </p>
|
3,422,095 | <p>I have been playing with Maclaurin series lately, I have been able to come across this:</p>
<p><span class="math-container">$\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5...$</span></p>
<p><span class="math-container">$\dfrac{1}{(1+x)^2}=1-2x+3x^2-4x^3+5x^4-6x^5+7x^6...$</span></p>
<p>I found out by accident that:</p>
<p><... | user21820 | 21,820 | <p>The most efficient way to obtain such formulae is to compute the <a href="http://math.stackexchange.com/a/2319671/21820">Newton series for <span class="math-container">$k$</span>-th powers</a>, and then use the fact that <span class="math-container">$1/(1-x)^{k+1} = \sum_{n=0}^∞ \binom{n+k}{k} x^n$</span> for <span ... |
23,003 | <p>I'm trying to illustrate the solutions to a textbook problem dealing with quadratic functions.</p>
<p>This will involve plotting a quadratic and overlaying the plot and the image.</p>
<p>Here is the textbook scan.....<img src="https://i.stack.imgur.com/La8Zs.jpg" alt="enter image description here"></p>
<p>The ide... | J. M.'s persistent exhaustion | 50 | <p>Try this:</p>
<pre><code>(* clip white borders *)
img = ImageCrop[Import["http://i.stack.imgur.com/La8Zs.jpg"]];
Plot[-14/81 x (x - 18), {x, -2, 19},
PlotRange -> {-2, 16}, PlotStyle -> Directive[Red, Thick, Dashed],
Prolog -> {Texture[img],
Polygon[{Scaled[{0, 0}], Scaled[{1, 0... |
23,003 | <p>I'm trying to illustrate the solutions to a textbook problem dealing with quadratic functions.</p>
<p>This will involve plotting a quadratic and overlaying the plot and the image.</p>
<p>Here is the textbook scan.....<img src="https://i.stack.imgur.com/La8Zs.jpg" alt="enter image description here"></p>
<p>The ide... | BoLe | 6,555 | <p>I enlarge the image canvas if throw flies outside.</p>
<pre><code>together[i, {4, 22, 10}]
</code></pre>
<p><img src="https://i.stack.imgur.com/e1kea.png" alt="Throw 1"></p>
<p>Parabola with zeroes <code>x1</code>, <code>x2</code> and vertex at <code>{(x1 + x2)/2, y3}</code>:</p>
<pre><code>y[{x1_, x2_, y3_}, x_... |
4,204,053 | <p>For any <span class="math-container">$t>0$</span> suppose that <span class="math-container">$f_t$</span> is a continuous function on <span class="math-container">$\mathbb{R}$</span> and uniformly bounded in <span class="math-container">$t$</span> : <span class="math-container">$\|f_t\|_\infty \leq C$</span>. Supp... | David C. Ullrich | 248,223 | <p>The Riemann-Lebesgue Lemma shows that <span class="math-container">$$f_t(x)=1+\sin(x/t)$$</span>is a counterexample.</p>
|
861,230 | <p>How to calculate:
$$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$
I thought I should find the integral on the path
$[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$.</p>
<p>I can easily take the residue in $i$
$$
Res_{z=i} \frac{x \sin(x)}{x^2+1}
\quad = \quad
\frac{i (e^{ii}-e^{-ii})}{2i}
\quad = \quad
\frac{i... | Koenraad van Duin | 88,135 | <p>I will share my work with you and hope that you can tell me if I understood everything well. I used two paths:</p>
<p>$$
\left\{ \begin{array}{ll}
A \quad = \quad [-R,R] \ \cup \ \{Re^{i\phi} \ : \ 0 \leq \phi \leq \pi\} \\
B \quad = \quad[-R,R] \ \cup \ \{Re^{i\phi} \ : \ \pi \leq \phi \leq 2\... |
123,494 | <p>In <em>Mathematica</em>, almost everything is notebook: your "Untitled-1.nb" is a notebook, Help documentation are a series of notebooks, even those windows helping you to draw things or format your notebooks are, themselves, notebooks.</p>
<p>But I occasionally find some exceptions and I want to know what are they... | Kuba | 5,478 | <p>With addition to Alexey's answer, <em>FrontEnd</em> probably hides them with:</p>
<pre><code> FrontEndExecute @ FrontEnd`SetNotebookInList[EvaluationNotebook[], False]
</code></pre>
<p>You can use it too to toggle appearance on <code>Notebooks[]</code> list.</p>
|
43,231 | <p>I'm trying to see why my textbook's solution is correct and mine isn't.</p>
<p>"Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$</p>
<p>First, the textbook's solution, which I understand and agree with fully:
<img src="https://i.stack.imgur.com/uyDlo.png... | Jack Schmidt | 583 | <p>Fractions can be written multiple ways.</p>
<p>$$\frac{-x-3}{(5y-4)} = \frac{x+3}{4-5y}$$</p>
<p>In general $$\frac{a-b}{c-d}=\frac{b-a}{d-c}$$</p>
<p>This is just multiplying both the top and the bottom by −1. In other words, your answer and the books differ by multiplication of $$\frac{-1}{-1} = 1$$</p>
|
361,045 | <p>I'm trying to prove that <span class="math-container">$\operatorname{Aut}(\mathbb Z_8)$</span> is isomorphic to <span class="math-container">$\mathbb Z_2 \oplus\mathbb Z_2$</span>, but I have no idea how to prove it. First of all, I'm trying to prove that <span class="math-container">$\operatorname{Aut}(\mathbb Z_8)... | Ittay Weiss | 30,953 | <p>You're on to a good start. An automorphism of a cyclic group is uniquely and completely determined by the image of any fixed generator, and that image must itself be a generator. That proves to you indeed that $Aut(\mathbb Z_8)$ has four elements. </p>
<p>Now, to go on, you just need to distinguish between the two ... |
4,427,446 | <p>There is a binary number of length <span class="math-container">$N$</span> which consists of a consecutive series of 1s. For example, if <span class="math-container">$N=5$</span> the number is <span class="math-container">$11111$</span>. How many ways are there to intervene on this number (i.e., replacing <span clas... | paw88789 | 147,810 | <p>Easier to count cases that we are not interested in. (I.e., sequences with at most <span class="math-container">$1$</span> uninterrupted block of <span class="math-container">$1$</span>s.)</p>
<p>There is one sequence of all <span class="math-container">$0$</span>s.</p>
<p>There are <span class="math-container">$n$<... |
2,654,507 | <blockquote>
<p>Find the residue of $\dfrac{z^2}{(z-1)(z-2)(z-3)}$ at $\infty$.</p>
</blockquote>
<p>We know that $\text{Res} (f)_\infty +\text{Res} (f)_{\text{ at other poles}}=0$</p>
<p>Now $f$ has poles at $1,2,3$ of order $1$.</p>
<p>Sum of residues of $f$ at $1,2,3=\dfrac{1}{2}+(-4)+\dfrac{9}{2}=1\implies \t... | Tiago Emilio Siller | 526,875 | <p>$(a^2+b^2)^3 = a^6 + 3a^4b^2 + 3a^2b^2 + b^6 \Rightarrow$</p>
<p>$ a^6+b^6 = (a^2+b^2)^3 - 3a^2b^2(a^2+b^2)$</p>
<p>Substituting $a=\sin(x)$ and $b=\cos(x)$:</p>
<p>$ \sin^6(x)+\cos^6(x) = (\sin^2(x)+\cos^2(x))^3 - 3\sin^2(x)\cos^2(x)(\sin^2(x)+\cos^2(x)) = \\
1 - 3\sin^2(x)\cos^2(x) \Rightarrow$</p>
<p>$ 16\sin... |
3,707,182 | <p>I just got done with an exam and one question was to determine the possible minimal polynomials of <span class="math-container">$A$</span>, if <span class="math-container">$A^3$</span> is the identity matrix. Note that <span class="math-container">$A$</span> is just some square matrix over <span class="math-containe... | Arthur | 15,500 | <p>You're right that <span class="math-container">$A$</span> fulfills <span class="math-container">$x^3-1$</span>, and the minimal polynomial must therefore be a factor of <span class="math-container">$x^3-1$</span>. You are missing two things, however:</p>
<ol>
<li>The minimal polynomial could also be <span class="ma... |
74,036 | <p>I have a 3D model of a heart in Mathematica and I'm trying to create a plane so that the open surface (as seen in the image below) is cut off so that the heart can have a solid, level surface. How can I combine this plane with my 3D contour plot?</p>
<pre><code>heart = (2 x^3 + y^2 + z^2 - 1)^3 - (1/10) x^2 z^3 - ... | halirutan | 187 | <p>Will this help you?</p>
<pre><code>c1 = ContourPlot3D[{heart == 0}, {x, -1.5, 1.5}, {y, -1.5, 1.5}, {z, -1.5, 1.5},
Mesh -> None, ContourStyle -> Opacity[0.8, Red],
RegionFunction -> Function[{x, y, z}, x > -0.3]];
c2 = ContourPlot3D[x == -.3, {x, -1.5, 1.5}, {y, -1.5, 1.5}, {z, -1.5, 1.5},
Mesh ... |
98,402 | <blockquote>
<p><strong>Theorem :</strong></p>
<p>If an odd number <span class="math-container">$n$</span> , <span class="math-container">$n > 1$</span> can be uniquely expressed as : <span class="math-container">$n=x^2-y^2$</span> ; <span class="math-container">$x,y \in \mathbb{Z}^{*}$</span> then</p>
<p><span clas... | Dan Brumleve | 1,284 | <p>If $p$ is of the form $4n+1$, you can skip odd values of $x$; and if $p$ is of the form $4n+3$, you can skip even values of $x$. So you can initialize $x$ to $0$ or $1$ depending on $p$ and always advance $x$ by $2$. This would seem to be an equivalent program which runs twice as fast. I believe I understand the ... |
1,863,151 | <p>I've seen this exercise in several statistics text, but how they get to the final formula is something that I don't quite get. How do two squared terms suddenly become a binomial term? I've been trying to figure out how to get to the final formula but I don't get anywhere near. Hope you can help me.</p>
<p><a href=... | Olivier Oloa | 118,798 | <p>One may recall that
$$
\bar{x}=\frac1N\cdot \sum_{n=1}^Nx_n
$$ giving
$$
\begin{align}
N^2\cdot \frac1N\sum_{n=1}^N(x_n-\bar{x})^2&=N\sum_{n=1}^N(x_n^2-2x_n\cdot\bar{x}+\bar{x}^2)
\\\\&=N\sum_{n=1}^Nx_n^2-2\:N\cdot\bar{x}\sum_{n=1}^Nx_n+N\cdot N\cdot \bar{x}^2
\\\\&=N\sum_{n=1}^Nx_n^2-2\:\left(\sum_{n=1}... |
34,600 | <p>Searching by <a href="/questions/tagged/closest" class="post-tag" title="show questions tagged 'closest'" rel="tag">closest</a> and <a href="/questions/tagged/position" class="post-tag" title="show questions tagged 'position'" rel="tag">position</a> I wasn't able to find an answer—but found <a ... | kglr | 125 | <p>In versions 10+, you can have a <code>NearestFunction</code> that returns multiple properties, such as the <code>"Index"</code> of the nearest element and the <code>"Element"</code> itself:</p>
<pre><code>SeedRandom[42];
haystack = RandomReal[1, 1000000];
f = Nearest[haystack ->{"Index", "Element"}];
AbsoluteTi... |
1,530,118 | <p>I was wondering if anyone could help with this $\epsilon–\delta$ definition of a limit. I have looked it up in my calculus book and online and I just don't understand how to do it.</p>
<p>Prove, using the $\epsilon–\delta$ definition of a limit that</p>
<p>$$\lim_{(x,y)\to(0,0)}\frac{(x^2-y)}{(4x^2+y^2)}$$</p>
| egreg | 62,967 | <p>If you can't yet use Taylor expansions, you can still use l'Hôpital with some simplifications to begin with.</p>
<h2>Edited question</h2>
<p>For the question where $e^{2x^4}$ is in the denominator instead of $e^{2x}$, the first step to do is the substitution $t=\sqrt[4]{x}$, noting that the function is even; so su... |
2,961,864 | <p><strong>Problem</strong></p>
<p>Prove using Stokes' theorem that
<span class="math-container">$$\int_C y dx +z dy + x dz = \pi a^2 3^.5,$$</span> where <span class="math-container">$C$</span> is the curve of intersection of the sphere <span class="math-container">$x^2+y^2+z^2=a^2$</span> and the plane <span class=... | B. Goddard | 362,009 | <p>You don't need to work out what <span class="math-container">$dS$</span> is. The curl of <span class="math-container">$F$</span> is <span class="math-container">$\langle -1,-1,-1\rangle$</span>, and the unit normal vector is the same thing, but divided by <span class="math-container">$\sqrt{3}$</span>, so your surf... |
64,392 | <p>thank you for reading my question.</p>
<p>I have a problem trying to export mixed data from Mathematica. I have different matrices and vectors, which should be combined to a output file.</p>
<p>Here a minimal-working-example:</p>
<pre><code>a = {{1}, {2}, {3}};
b = {{4}, {5}, {6}};
c = {7, 8, 9};
d = Transpose[{a... | Jan | 21,763 | <p>The lists "a" and "b" are nested lists. With the <a href="http://reference.wolfram.com/language/ref/Flatten.html" rel="nofollow noreferrer">Flatten</a> option, the nested lists my be flattened and the nesting removed.</p>
<p>The answer is, as posted from <a href="https://mathematica.stackexchange.com/users/1356/%C3... |
3,098 | <p>This is a really newbie question, but it has me confused. Why does this code <strong>work without</strong> <code>// MatrixForm</code> and <strong>doesn't work with</strong> <code>// MatrixForm</code>?</p>
<pre><code>cov = {{0.02, -0.01}, {-0.01, 0.04}} // MatrixForm
W = {w1, w2}; FindMinimum[ W.cov.W, W]
</code></p... | Verbeia | 8 | <p>David's answer is correct and the one you need to solve your specific problem. I thought nonetheless that it is worth providing some additional information that might help explain how to diagnose similar issues.</p>
<p>Matrix/tensor operations like <code>Dot</code> and <code>Inverse</code> are designed to work with... |
2,144,481 | <p>Let us consider the sequence $(a_n)_{n \ge 1}$ such that
$$a_n=\frac {1}{\sqrt {n^2+1}}+ \frac {1}{\sqrt {n^2+2}} + \dots +\frac {1}{\sqrt {n^2+n}}.$$ Show that for every $k \in \Bbb N, k\gt 0,$ we have $a_n \ge a_k$, for every $n \ge k^2$.</p>
<p>The only method I know is computing the difference $a_{n+1}-a_n$, ... | HGE | 415,588 | <p>Notice that for $k=1$ and $n=1$, $n\geq k^2$ but $a_n=a_k$ and thus $a_n>a_k$ does not hold for $k \in \Bbb N, k\gt 0$.</p>
|
2,223,267 | <p>For<br>
$$e^{-j\pi n}$$</p>
<p>How does this become
$$(-1)^n$$</p>
<p>or is it actually
$$(-1)^{-n}$$
I have checked on calculator and values are all the same when the same n value is used</p>
| Saketh Malyala | 250,220 | <p>$e^{πj}=-1$</p>
<p>So $e^{-πjn}=(e^{πj})^{-n}=(-1)^{-n}$.</p>
<p>I am assuming $j$ is the imaginary unit, or $\sqrt{-1}$. </p>
|
264,025 | <p>Suppose we start with a $n\times n$ matrix with entries sampled independently and uniformly at random from $[0,1]$. The weight of a set of entries will simply be the sum of those entries. A permutation refers to a set of $n$ entries, no two on the same row or column.</p>
<p>Pick a permutation whose corresponding en... | Brendan McKay | 9,025 | <p>This is a good question and I'm fairly sure the answer is not known. My collection of non-transitive cubic connected walk-regular graphs looks like this:</p>
<pre><code> 1 graphs : n=20; girth=6; bipartite; radius=4; diameter=5; orbits=2
1 graphs : n=30; girth=6; bipartite; radius=5; diameter=5; orbits=3
1 gra... |
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| Robert Israel | 8,508 | <p>Expanding out the left side of the first equation,</p>
<p>$$ \frac{4 \tan \alpha - 4 \tan \beta}{1 + \tan \alpha \tan \beta} = 3 \tan \alpha$$</p>
<p>Thus</p>
<p>$$ \tan \beta = \frac{\tan \alpha}{4 + 3 \tan^2 \alpha}$$</p>
<p>Writing $\tan \alpha = \dfrac{\sin \alpha}{\cos \alpha}$, this becomes</p>
<p>$$ \tan... |
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| Dennis Gulko | 6,948 | <p>Recall that $$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$$
Hence $$\tan\alpha-4\tan\beta=3\tan^2\alpha\tan\beta \hspace{8pt}\Rightarrow\hspace{8pt}\tan\alpha=(3\tan^2\alpha+4)\tan\beta\hspace{8pt}\Rightarrow$$
$$\begin{align*}\tan\beta=&\frac{\tan\alpha}{3\tan^2\alpha+4}=\frac{\frac{\s... |
208,694 | <p>Could someone explain how to correctly prove that $$\lim_{n\to\infty}\sin\frac{1}{n}$$ where $n=1,2,\cdots,n$ doesn't exist.
I have no problem with it if $\sin\frac{1}{x}$ where $x$ is real, because just taking values $x=\frac{2}{(2n-1)\pi}, x=\frac{1}{n\pi}, x=\frac{2}{(2n+1)\pi}$ it is clear, for example, by Cauch... | Sean Eberhard | 23,805 | <p>From your discussion it seems likely that you actually meant to ask about the limit of $\sin n$. Here is a possible proof that $\sin n$ does not converge. Since</p>
<p>$$ \sin(n+1) = \sin(n)\cos(1) + \cos(n)\sin(1),$$</p>
<p>it follows that if $\sin n$ converges so does $\cos n$, and therefore $e^{in} = \cos n + i... |
208,694 | <p>Could someone explain how to correctly prove that $$\lim_{n\to\infty}\sin\frac{1}{n}$$ where $n=1,2,\cdots,n$ doesn't exist.
I have no problem with it if $\sin\frac{1}{x}$ where $x$ is real, because just taking values $x=\frac{2}{(2n-1)\pi}, x=\frac{1}{n\pi}, x=\frac{2}{(2n+1)\pi}$ it is clear, for example, by Cauch... | Mikko Korhonen | 17,384 | <p>Using sequences, we can characterize continuity at a point for real functions.</p>
<p>Function $f: A \rightarrow \mathbb{R}$ is continuous at a point $x_0 \in A$ if and only if for any sequence $(x_n)$ in $A$ converging to $x_0$, we have $$\lim_{n\to\infty}f(x_n) = f\left(\lim_{n\to\infty}x_n\right) = f(x_0)$$</p>
... |
19,590 | <p><a href="http://www.xamuel.com/graphs-of-implicit-equations/" rel="noreferrer">Here</a> are several equations, it seems that Mathematica couldn't plot them well, although I set PlotPoints>100</p>
<pre><code> ContourPlot[Csc[1. - x^2] Cot[2. - y^2] - x*y == 0,
{x, -10, 10}, {y, -10, 10}, PlotPoints -> 120]
</... | Xerxes | 5,406 | <p>OK, I think based on the number of times I crashed my kernel trying to reproduce that plot that brute force is not the right answer. Here's a technique that's a little more clever:</p>
<ol>
<li>Select a random point in the region of interest.</li>
<li>Use a solver (steepest descent, say) to get onto a zero contour.... |
1,790,032 | <p>Say you have a number $x^{\sqrt 2}$.</p>
<p>Is there any way to represent this number so that there's no root (or irrational) as the exponent (so that it's easier to understand for me)? I just can't wrap my head around this.</p>
<p>I was thinking something like $$x^{2^{1/2}}$$ and extending on that idea, but I don... | user21820 | 21,820 | <p>I guess you can't wrap your head around an irrational exponent because you don't know what it really is supposed to <strong>mean</strong>, and you're not just asking how to <strong>compute</strong> it. That's an excellent question, and here is the answer.</p>
<h2>Integer exponents</h2>
<p>The basic idea is that expo... |
1,790,032 | <p>Say you have a number $x^{\sqrt 2}$.</p>
<p>Is there any way to represent this number so that there's no root (or irrational) as the exponent (so that it's easier to understand for me)? I just can't wrap my head around this.</p>
<p>I was thinking something like $$x^{2^{1/2}}$$ and extending on that idea, but I don... | MathematicianByMistake | 237,785 | <p>I am not sure you will find this more to your liking, but you can use the fact that $$\cos\frac{\pi}{4}=\frac{\sqrt2}{2}$$</p>
<p>to obtain $$x^\sqrt2=x^{2\cos\frac{\pi}{4}}$$</p>
<p>Of course there is still an irrational hidden there, but a geometric interpretation might help your intuition.</p>
|
3,625,000 | <p>I have read that an n-dimensional cube has <span class="math-container">$2^n$</span> vertices, but I can't find a proof for that. What is the explanation to why that's true?</p>
| Community | -1 | <p>You can define an interval, a square, a cube, a tesseract, an hypercube… by its vertices</p>
<p><span class="math-container">$$(0),(1)$$</span></p>
<p><span class="math-container">$$(0,0), (0,1), (1,0), (1,1)$$</span></p>
<p><span class="math-container">$$(0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0, 0), (1,1,0), (1,... |
136,121 | <p>I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?</p>
| François G. Dorais | 2,000 | <p>There are many algorithms that are suitable for different contexts. Asymptotically, a combination of Newton's method with FFT is the best known method according to R. P. Brent <em>Multiple-precision zero-finding methods and the complexity of elementary function evaluation</em> [Analytic computational complexity (Pro... |
49,015 | <p>This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and t... | Joël | 9,317 | <p>No.</p>
<p>That is perhaps a little too categorical, but a mathscinet search with Grothendieck as author and "modular form" or "forme modulaire" as "anywhere" gives no result. I don't remember him mentionning modular forms in "Recoltes et Semailles" either.</p>
<p>More to the point, it is a commonplace in the fiel... |
1,641,589 | <p>I have a theorem for the hamming bound or the sphere packing bound.</p>
<p>A q-ary $(n, m, 2e+1)$ code satisfies
$$M \bigg\{ \binom {n}{0} + \binom{n}{1} (q-1)+...+\binom{n}{e}(q-1)^e\bigg\} \leq q^n $$</p>
<p>What is $q^n$?</p>
<p>What is this theorem trying to say?</p>
| xxxxxxxxx | 252,194 | <p>You have a $q$-ary $(n,m,2e+1)$ code, so you should know what $q$ and $n$ represent ($q$ is the size of the alphabet, $n$ is the length of the codewords, $m$ is the number of codewords, and $2e+1$ is the minimum distance between codewords). So $q^n$ is just this value, $q$ to the $n$th power.</p>
<p>You are using ... |
615,614 | <h1>Question</h1>
<p>Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work?</p>
<h1>My thoughts</h1>
<h2>Integral exponents</h2>
<p>Defining $A^k$ for $k\in\mathbb N$ is easy in terms of repeated multi... | MvG | 35,416 | <p>As <a href="https://math.stackexchange.com/users/59101/tom">@tom</a> pointed out in a comment, the power of a matrix can be defined in terms of <a href="http://en.wikipedia.org/wiki/Logarithm_of_a_matrix" rel="nofollow noreferrer">logarithm of a matrix</a> and <a href="http://en.wikipedia.org/wiki/Matrix_exponential... |
615,614 | <h1>Question</h1>
<p>Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work?</p>
<h1>My thoughts</h1>
<h2>Integral exponents</h2>
<p>Defining $A^k$ for $k\in\mathbb N$ is easy in terms of repeated multi... | ccampisano | 633,046 | <p>I came to the need/wish to extend <span class="math-container">$A^n$</span> to <span class="math-container">$A^z$</span> from a geometrical perspective (where <span class="math-container">$n$</span> is an integer number and <span class="math-container">$z$</span> real one), so I hope this can help, by providing a ge... |
878,237 | <p>Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive order only once.</p>
<p>Now we need to flip cards in such a way that sum of number of upper face of cards is maximum.<... | Mathsource | 12,624 | <p>$$
(x + y)^n = \sum_{k=0}^{n}{n \choose k}x^ky^{n-k} \quad \Rightarrow \quad (2x^2 + 3)^7 = \sum_{k=0}^{7}{7 \choose k}(2x^2)^{7-k}3^k
$$
Like this, $2(7 - k) = 10 \ \Rightarrow \ k =2$. Therefore, the term is ${7 \choose 2}32\cdot 9x^{10}$.</p>
|
878,237 | <p>Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive order only once.</p>
<p>Now we need to flip cards in such a way that sum of number of upper face of cards is maximum.<... | user421443 | 421,443 | <p>$r$ must be equal to $5$, so the term that will have the exponent $10$ is the $6_{th}$ term..</p>
<p>since to get $r$, </p>
<p>(term desired) - 1 </p>
<p>or</p>
<p>$t-1=r$</p>
|
215,774 | <p>Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The following two elements are obviously in $G$: the function $t$ (translation) where $t(x)=x+1$ for each $x \in \mathbb... | Hagen von Eitzen | 39,174 | <p>I assume, you have also noted that $i\in\{0,1\}$ when writing an element as $r^it^j$.
Essentially, there are just four cases to consider:</p>
<ul>
<li>$t^j \cdot t^k$: Clearly, this is $t^{j+k}$.</li>
<li>$rt^j \cdot t^k$: Clearly, this is $rt^{j+k}$.</li>
<li>$t^j \cdot rt^k$: Note that $(t^j rt^k)(x) = (t^j r)(x+... |
3,643,186 | <p>A <span class="math-container">$k^{th}$</span> order PDE is defined by <span class="math-container">$$F(D^ku(x),D^{k-1}u(x),\dotsc,Du(x),u(x),x)=0,$$</span> where <span class="math-container">$x$</span> is an element of <span class="math-container">$U$</span>. I know that <span class="math-container">$\mathbb{R}^n$... | Arctic Char | 629,362 | <p>Let me just recall the notations: </p>
<ul>
<li><span class="math-container">$U$</span>: an open set in <span class="math-container">$\mathbb R^n$</span>, </li>
<li><span class="math-container">$x = (x_1, \cdots, x_n)$</span> is in <span class="math-container">$U$</span>, </li>
<li><span class="math-container">$u$<... |
2,026,143 | <p>What is need to be proven, in a proof by contradiction that a set is closed?</p>
<p>If we have to show that $K$ is closed that mean that we need to show that $K^{C}$ is open.
Let there be $x\in K^{C}$ we need to show that $B(x,\delta)\subset K^{C}$</p>
<p>What is the contrary assumption?</p>
<p>For all $\delta>... | fleablood | 280,126 | <p>"What is need to be proven, in a proof by contradiction that a set is closed?"</p>
<p>There are two possible approaches depending on your definition of "closed":</p>
<p>1) $K$ is closed if $K^c$ is open.</p>
<p>So a proof by contradiction is to assume $K^c$ is not open.</p>
<p>The definition of "open" is: For al... |
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | user1337 | 62,839 | <p>We know that $(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k$.Differentiate this, to get $n(1+x)^{n-1}=\sum_{k=0}^n \binom{n}{k} k x^{k-1}$. Multiply by $x$ to get $nx(1+x)^{n-1}=\sum_{k=0}^n \binom{n}{k} k x^k$. Take $x=1$ to get the first sum, And repeat this process for sums involving higher powers of $k$.</p>
|
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Marko Riedel | 44,883 | <p>Suppose we seek to evaluate
$$\sum_{k=0}^n {n\choose k} k^p.$$</p>
<p>Introduce
$$k^p = \frac{p!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{p+1}} \exp(kz) \; dz.$$</p>
<p>This yields for the sum
$$\frac{p!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{p+1}}
\sum_{k=0}^n {n\choose k} \exp(kz) \; dz
\\ = \frac{p!}{2\pi i}... |
214,007 | <p>I have an experimental data set: </p>
<pre><code>data1 = {{71.6`, 0.41`}, {27.2`, 4.96`}, {59.3`, 0.18`}, {46.`,2.72`},
{42.2`, 1.06`}, {89.1`, 3.75`}, {88.6`, 1.9`}, {62.3`,1.8`},
{35.5`,1.84`}}
</code></pre>
<p>In order to eliminate unrealistic data points step by step automatically, I fit and ... | MelaGo | 63,360 | <p>Simplified your code a bit:</p>
<pre><code>data1 = {{71.6`, 0.41`}, {27.2`, 4.96`}, {59.3`, 0.18`}, {46.`, 2.72`}, {42.2`, 1.06`}, {89.1`, 3.75`}, {88.6`, 1.9`}, {62.3`, 1.8`}, {35.5`, 1.84`}}
lm = Fit[data1, {1, x}, x]
(* 2.87217 - 0.0138549 x *)
Show[
ListPlot[data1, PlotStyle -> PointSize[.02], PlotRange ... |
4,616,445 | <p>Could anyone enlighten me on how to go about expanding the following function around <span class="math-container">$x_0 = 0$</span>:</p>
<p><span class="math-container">$$
f(x):= \log(1+x)e^{x}
$$</span></p>
<p>I have tried using Cauchy Product Series and bruteforce computation of the coefficients but I always find m... | Dark Malthorp | 532,432 | <p>Note that <span class="math-container">$$
\frac{d^n}{dx^n} \log(1+x) = (-1)^{n-1}\frac{(n-1)!}{(1+x)^n}
$$</span>
thus <span class="math-container">$$
f^{(n)}(x) = \log(1+x)e^x + \sum_{k=1}^n \binom{n}{k}(-1)^{k-1} \frac{(k-1)!}{(1+x)^n} e^x
$$</span>
Evaluating at <span class="math-container">$0$</span> gives the <... |
108,404 | <p>Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. </p>
<p>In which cases the conjecture is known to be true?</p>
| Alexander Chervov | 10,446 | <p>From <a href="http://www2.math.technion.ac.il/~gelaki/research_summary.pdf">Shlomo Gelaki research statement</a> (which is nice survey, by the way):</p>
<blockquote>
<p>We also proved that the dimension of
an irreducible representation of a
semisimple Hopf algebra H, which is
either quasitriangular or
cot... |
108,404 | <p>Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. </p>
<p>In which cases the conjecture is known to be true?</p>
| Julian Kuelshammer | 15,887 | <p>There is a new survey on Kaplansky's sixth conjecture by L. Dai and J. Dong, available on the <a href="http://arxiv.org/abs/1409.2545" rel="noreferrer">arxiv</a>. Among other results, it mentions the following (always assuming $\operatorname{char} k=0$):</p>
<p>Special primes:</p>
<blockquote>
<p>If a semisimple... |
776,615 | <p>Consider a simple function $f(x,y)=\frac{x}{y}, x,y \in (0,1]$, the Hessian is not positive semi definite and hence it is a non convex function. However, when we plot the function using Matlab/Maxima, it "appears" convex. For the sake of clarity we want to find points which violate the definition of convexity, in ot... | Michael Grant | 52,878 | <p>The Hessian is $$\nabla^2 f(x) = \begin{bmatrix} 0 & -y^{-2} \\ -y^{-2} & 2xy^{-3} \end{bmatrix}=\frac{1}{y^3}\begin{bmatrix} 0 & -y \\ -y & 2x\end{bmatrix}.$$
For nonzero $y$, the determinant $-y^{-4}$ is strictly negative. This means that there must be exactly one positive eigenvalue and one negati... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.