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 |
|---|---|---|---|---|
650,866 | <p>So I have the function $$ e^{-2x} $$ and if I derive this I thought that I should get $$ -2xe^{-2x} $$ But the $x$ disappears, why? Is it an inner derivative and because of that, I also have to differentiate the expression $-2x$ when I put it in front of $e$? If that is the case, then $x$ would be 1 and -2 is the on... | Glen O | 67,842 | <p>Go back to the basic definition of derivative. You have this:</p>
<p>\begin{align}
\frac{d}{dx}\left(e^{-2x}\right) &= \lim_{h\to0} \frac{e^{-2(x+h)}-e^{-2x}}{h}\\
&=\lim_{h\to0} \frac{e^{-2x}(e^{-2h}-1)}{h}\\
&=e^{-2x}\lim_{h\to0} \frac{e^{-2h}-1}{h}
\end{align}
As you can see, there is no $x$ outside ... |
2,419,485 | <blockquote>
<p>In a certain family four girls take turns at washing dishes. Out of a total of four breakages, three were caused by the youngest girl, and she was thereafter called clumsy. Was she justified in attributing the frequency of her breakages to chance?</p>
</blockquote>
<p>I'm not sure how to solve the fo... | Ilia Vatahov | 761,820 | <p>There is a mistake in the book and it's not just a typo. It seems that the provided answers are for another task.</p>
<ul>
<li><p>A: three <strong>or</strong> four breakages are caused by one girl.</p>
</li>
<li><p>B: three <strong>or</strong> four breakages are caused by the youngest girl.</p>
</li>
</ul>
<p>If we ... |
479,594 | <p>I was wandering which is the best way to generate various combinations of $x_i$ such that $$\sum\limits_{i=1}^7 x_i = 1.0$$</p>
<p>where $ x_i \in \{0.0, 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$</p>
<p>I can generate these using brute-force, i.e checking through all $ 11^7$ combinations and only taking those whi... | obataku | 54,050 | <p>By inspection we determine a particular solution to $2a_n-a_{n-1}=1/2$ is given by $a_n=1/2$ trivially -- try an ansatz of the form $a_n=k$ and thus we get $k=2k-k=1/2$.</p>
<p>Considering the homogeneous case, $2a_n-a_{n-1}=0$, let $a_n=\lambda^n$ hence:$$2\lambda^n-\lambda^{n-1}=0\\2\lambda-1=0\\\lambda=\frac12$$... |
1,259,853 | <p>Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)?</p>
<p>I have tried to applied the chain rule, but it comes something completely different:</p>
<p>$$\frac{1}{n} n^{\frac{1}{n} - 1} \cdot 1 = \frac{1}{n} n^\fr... | robjohn | 13,854 | <p>Write
$$
n^{1/n}=\exp\left(\frac1n\log(n)\right)
$$
Then the chain rule, followed by the product rule, says
$$
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}n}n^{1/n}
&=\exp\left(\frac1n\log(n)\right)\frac{\mathrm{d}}{\mathrm{d}n}\left(\frac1n\log(n)\right)\\
&=n^{1/n}\left(\frac1{n^2}-\frac{\log(n)}{n^2}\right)... |
894,476 | <p>I don't have a strong background in probability/statistics and I'm trying to understand the example at <a href="http://rationalwiki.org/wiki/Extraordinary_claims_require_extraordinary_evidence#Probability_theory" rel="nofollow">http://rationalwiki.org/wiki/Extraordinary_claims_require_extraordinary_evidence#Probabil... | André Nicolas | 6,312 | <p>You can apply it to Simpson's Rule. However, Simpson's Rule is obtained by one extrapolation step from the Trapezoidal Rule, so it makes no difference. </p>
|
201,999 | <p>Prove that the given sequence ${a_n}$ diverges to infinity.</p>
<p>$a_n=\frac{n^3+5}{-n^2+8n}$</p>
<p>I believe that the sequence diverges to -infinity. And I have this for my proof so far:</p>
<p>Let $M>0$ and let $N=$ ?. Then $n>N$ implies... I am confused on how to solve for the N. I believe I have to ma... | Community | -1 | <p>Working directly from the definition, you're trying to show that for $n$ sufficiently large, you can make $a_n < -M$ for any $M \in \mathbb{N} .$ It helps to rewrite the general term as $\frac{n + 5/n^2}{-1 + 8/n}.$ The end behavior is suggested by the leading terms in the numerator and the denominator, so there ... |
38,439 | <p>I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue integral, and then we switch framework to manifolds, and we have that trick of using partitions of unity to define inte... | Dmitri Pavlov | 402 | <p>Here is my own favorite construction of the (Lebesgue) integral.</p>
<p>Suppose M is an arbitrary smooth manifold.
Denote by Or(M) the orientation line bundle of M.
This bundle is equipped with a canonical Riemannian metric.
Vectors of length 1 in the fiber of Or(M) over a point p∈M correspond canonically to the tw... |
3,493,519 | <p>Can I get a verification if this is the right way to approach this problem?</p>
<blockquote>
<p>Give an example of a linear map <span class="math-container">$T$</span> such that <span class="math-container">$\dim(\operatorname{null}T) = 3$</span> and <span class="math-container">$\dim(\operatorname{range}T) = 2$<... | Arthur | 15,500 | <p>This is completely correct. This will give a linear map with the properties you're asked for.</p>
<p>I think that it is a bit too general to actually be "an example". I think it would be better if you <em>actually pick</em> a concrete basis. But that's a personal aesthetic belief, and one would have to be pretty pe... |
1,682,818 | <p>$L(G)$ is an undirected graph without parallel edges and loops such that:</p>
<p><strong>1. every edge in $G$ is an vertex in $L(G)$<br>
2. two vertices in $L(G)$ are connected by edge only if their edges in $G$ have a common vertex.</strong></p>
<p>The mission is to <strong>express the number of edges in $L(G)$ a... | Ove Ahlman | 222,450 | <p>You get $\binom{d(v)}2$ since you need to calculate the number of possible, unordered, pairs when you have $d(v)$ elements. Note especially that, if we call one of the edges in $v$ by the name $a$ then $a$ has $d(v)-1$ different other edges to be paired together with. Thus the answer $d(v)/2$ far from the truth.</p>... |
1,682,818 | <p>$L(G)$ is an undirected graph without parallel edges and loops such that:</p>
<p><strong>1. every edge in $G$ is an vertex in $L(G)$<br>
2. two vertices in $L(G)$ are connected by edge only if their edges in $G$ have a common vertex.</strong></p>
<p>The mission is to <strong>express the number of edges in $L(G)$ a... | Brian M. Scott | 12,042 | <p>Suppose that $G$ is the following graph:</p>
<pre><code> a
|
b
/ \
c d
</code></pre>
<p>It has edges $ab,bc$, and $bd$, so $L(G)$ has three vertices for which I will use the labels $v_... |
3,575,417 | <p>I have to prove the non-existence of a continuous bijection between <span class="math-container">$[0,1)$</span> and <span class="math-container">$\mathbb{R}$</span>.</p>
<p><strong>My attempt</strong>:</p>
<p>Since <span class="math-container">$\mathbb{R}$</span> is homeomorphic to <span class="math-container">$(0... | Community | -1 | <p>If a map <span class="math-container">$f:[0,1)\to (0,1)$</span> satifies the intermediate value property, then <span class="math-container">$f\left[(0,1)\right]=I$</span> is a subinterval of <span class="math-container">$(0,1)$</span>. If <span class="math-container">$f$</span> is bijective, then <span class="math-c... |
3,575,417 | <p>I have to prove the non-existence of a continuous bijection between <span class="math-container">$[0,1)$</span> and <span class="math-container">$\mathbb{R}$</span>.</p>
<p><strong>My attempt</strong>:</p>
<p>Since <span class="math-container">$\mathbb{R}$</span> is homeomorphic to <span class="math-container">$(0... | Henno Brandsma | 4,280 | <p>Suppose <span class="math-container">$f:[0,1) \to \Bbb R$</span> is a continuous bijection.
Then <span class="math-container">$f[(0,1)]=\Bbb R \setminus \{f(0)\}$</span> which is a contradiction, as <span class="math-container">$(0,1)$</span> is connected and so its continuous image is too, while <span class="math-... |
1,246,356 | <p>Let $A,B \in {M_n}$ . suppose $A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?</p>
| Prasun Biswas | 215,900 | <p>$$\sum_{k=5}^{\sqrt n}\frac{\ln(\ln k)}{k\ln k}\approx \int\limits_5^{\sqrt n}\frac{\ln(\ln k)}{k\ln k}\,\mathrm dk\stackrel{t=\ln k}=\int\limits_{\ln 5}^{0.5\ln n}\frac{\ln t}{t}\,\mathrm dt\stackrel{u=\ln t}=\int\limits_{\ln(\ln 5)}^{\ln(0.5\ln n)}u\,\mathrm du$$</p>
<p>Can you take it from here?</p>
|
3,454,725 | <p>I am reading real analysis book and encountered this symbol <span class="math-container">$\wedge$</span> and <span class="math-container">$\vee.$</span></p>
<p>The author says following:</p>
<ol>
<li><span class="math-container">$f\wedge g=\frac{1}{2}(f+g-
|f-g|)$</span>,</li>
<li><span class="math-container">$f\v... | fleablood | 280,126 | <p>The way I've seen it done is to specify <span class="math-container">$\{i_1, ....., i_k\}\subset \{1,...., |\phi_M|\}$</span> and <span class="math-container">$A = \{x_{i_j}|x_{i_j} \in \phi_M\}$</span>.</p>
<p>In fact just <span class="math-container">$A=\{x_{i_j}\}\subset \phi_M$</span> is usually understood that... |
601,951 | <p><em><strong>2</strong> + <strong>5</strong> + <strong>8</strong> + . . . + <strong>(6n-1)</strong> = <strong>n(6n+1</strong>)</em></p>
<p>This is what I have so far. </p>
<p>The <strong>sum</strong> of <strong>(3j-1)</strong> from <strong>j=1</strong> to <em>something I`m not sure of</em>.</p>
| ccorn | 75,794 | <p>If you are tired of trying to find an ingenious proof,
here is a computer-aided procedure for proving the identity.</p>
<p>The nomenclature follows that in <a href="http://www.math.upenn.edu/%7Ewilf/AeqB.pdf" rel="nofollow noreferrer">Petkovšek, Wilf, Zeilberger (1997): <span class="math-container">$A=B$</span></a>.... |
4,405,145 | <p>Given <span class="math-container">$x_1, x_2, x_3, x_4, x_5$</span> be independent standard normal random variable and <span class="math-container">$\bar x$</span> the sample mean <span class="math-container">$\bar x= (x_1 + x_2 + x_3 + x_4 + x_5)/5$</span>. Then <span class="math-container">$\Pr(\bar x\leqslant c)$... | user97357329 | 630,243 | <p><strong>A first solution by Cornel Ioan Valean</strong></p>
<p>One of the possible options here is to start by splitting the main integral and write
<span class="math-container">$$\int_0^1 \frac{\arctan^2(x)}{x}\log\left(\frac{x}{(1-x)^2}\right)\textrm{d}x$$</span>
<span class="math-container">$$=\underbrace{\int_0^... |
3,583,879 | <blockquote>
<p>a) $P_5=11$$</p>
<p>b) <span class="math-container">$P_1+P_2+P_3+P_4+P_5 =26$</span></p>
</blockquote>
<p>For the first part
<span class="math-container">$$\alpha^5+\beta ^5$$</span>
<span class="math-container">$$=(\alpha^3+\beta ^3)^2-2(\alpha \beta )^3$$</span></p>
<p>I found the value of <... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Use <span class="math-container">$$(a^3+b^3)(a^2+b^2)=a^5+b^5+a^2b^2(a+b)$$</span> as we know <span class="math-container">$a+b, ab$</span></p>
|
917,272 | <p>The following is an old exam problem (Calc III). It looks simple and technical, but I end up with a difficult integral and I guess I have a mistake somewhere.</p>
<p>We are given the vector field $F(x,y,z)=(4z+2xy,x^2+z^2,2yz+x)$. We are asked to calculate the line integral $\int_{C} \vec{F} \cdot d\vec{r}$, where ... | Andrés E. Caicedo | 462 | <p>Let $C_0=X$ and $C_{n+1}=f(C_n)$ for all $n$. By induction, note that $C_n\supseteq C_{n+1}$ for all $n$. Let $C=\bigcap_n C_n$, and note that $C$ is nonempty, being the intersection of a decreasing sequence of nonempty compact sets. </p>
<p>To check that $f(C)=C$, consider a point $x\in C$. Since $x\in C_{n+1}$ fo... |
917,272 | <p>The following is an old exam problem (Calc III). It looks simple and technical, but I end up with a difficult integral and I guess I have a mistake somewhere.</p>
<p>We are given the vector field $F(x,y,z)=(4z+2xy,x^2+z^2,2yz+x)$. We are asked to calculate the line integral $\int_{C} \vec{F} \cdot d\vec{r}$, where ... | Marc Bogaerts | 118,955 | <p>This sketch of a proof is based on the "special" definition of compactedness on metric spaces : <a href="http://en.wikipedia.org/wiki/Metric_space#Compact_spaces" rel="nofollow">compact metric spaces</a>. Take any point in $x\in X$ and consider the sequence $f(x), f(f(x)), \ldots$ then there is a subsequence that co... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | Mauro ALLEGRANZA | 108,274 | <p>In a proof with multiple <em>assumptions</em> you have to choose one of them to be "blamed" for the contradiction.</p>
<p>Think to your example in terms of <em>assumptions</em>; you start with a couple of them (they can be two <em>Lemmas</em> already proved, or two hypotheses) :</p>
<blockquote>
<p>$P→Q$ and $R→... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | David | 119,775 | <blockquote>
<p>So my question is: in general, when proving by contradiction, how do we know which assumption exactly is false? And how do we know that exactly one assumption must be wrong in order to proceed with the proof?</p>
</blockquote>
<p>If you make more than one assumption and get a contradiction, you canno... |
3,478,098 | <p><a href="https://i.stack.imgur.com/dcxhi.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dcxhi.jpg" alt="enter image description here" /></a></p>
<p>Guys, Why is this weird statement true? It seems counterintuitive to me I cannot understand or lack creativity understanding it can you help me expla... | David K | 139,123 | <p>It all comes from the three lines before the part you highlighted.</p>
<p>You might be thinking that no matter how small we make <span class="math-container">$\delta,$</span> there is always a rational number in the interval <span class="math-container">$(a - \delta, a + \delta).$</span>
If so, you would be right.
... |
462,397 | <p>So, I read the John Baez essay "Lectures on n-categories and cohomology" and I understand the notion of a (-1)-category" and a (-2)-category" and how to derive them. However, I'm not totally clear on what a (-1)-morphism is.</p>
<p>At nLab at <a href="http://ncatlab.org/nlab/show/k-morphism" rel="nofollow noreferr... | user062295 | 305,314 | <p>For an example, take the $\infty$ category $Kom$, i.e. the nerve of the category of chain complexes over a field. A $-1$ morphism is a map from one chain complex to another chain complex of degree $-1$.</p>
|
1,016,682 | <p>is my proof correct?</p>
<p>Definition:</p>
<p>Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the closure of X is denoted as $\overline X$ and is defined to be the set of all the adherent points of $X$.... | idm | 167,226 | <p>An other way to prove:
$\overline{\bar X}$ is the smallest close set that contain $\bar X$. But $\bar X$ is close then the smallest close set that contain $\bar X$ is $\bar X$, therefore $$\overline{\bar X}=\bar X$$</p>
|
86,536 | <p>Considering we have a an association:</p>
<pre><code>assc = <|"A" -> <|"a" -> 1, "aa" -> 2|>, "B" -> 0, "C" -> 5,"D" -> <|"d" -> 2, "dd" -> 12|>|>
</code></pre>
<p>Let's also consider we have 2 known lists and one list for nested keys
(**
- <em>how to create this list?</... | C. E. | 731 | <p>This solution will traverse all nested associations regardless of depth and replace each value with the value that corresponds to its key in <code>replacements</code>.</p>
<pre><code>f[Rule[key_, assoc_Association]] := Rule[key, AssociationMap[f, assoc]]
f[Rule[key_, val_]] := Rule[key, key /. replacements]
replac... |
3,300,469 | <p>I have a problem counting all the possible ways of "pairing" two datasets of size n and m, including partial pairing. </p>
<p>Example:
Assume we have two sets <span class="math-container">$\{A,B\}$</span> and <span class="math-container">$\{1,2,3\}$</span>. My aim is to find all ways of pairing letters with numbers... | SlipEternal | 156,808 | <p>Assume your sets are <span class="math-container">$A,B$</span> with <span class="math-container">$|A|\le |B|$</span> and <span class="math-container">$A\cap B = \emptyset$</span>. Then you are looking for the sum of the ways for each <span class="math-container">$X\subset A$</span>, the number of injections <span cl... |
1,515,823 | <p>I am doing my research in Functional Analysis, especially in "Generalized inverse of Linear Maps".</p>
<p>I have come across Probability by studying only the methods or Distributions(like Binomial, poisson, normal,etc)</p>
<p>Now I wish to study the mathematical background and intuitive way of looking on i... | Trajan | 119,537 | <p>I think your best bet would be <strong>"Probability And Random Processes" by Grimmett and Stirzacker</strong>, see <a href="http://www.amazon.co.uk/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220" rel="nofollow">http://www.amazon.co.uk/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220</a>. It... |
45,771 | <p>Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.</p>
<p>If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ and $(S(t),t \ge 0)$ is the Dirichlet heat semigroup on $L^p(U)$ then $(S(t) f)(x) = \int_U G_U(t,x,y) f(y)\,dy$... | Piero D'Ancona | 7,294 | <p>I would like to know a complete answer to this question myself, since this is useful in a number of situations. From what I know, contrary to estimates for the kernel which are a quite general phenomenon (see the classical book by Davies for your case), estimates for the derivatives are much more subtle and are conn... |
1,398,956 | <p>I saw from literature that the expected value of a random variable $f(X)$ is either $E f(X)$, $E(f(X))$ or $E[f(X)]$. Is there a standard which one notation should one use? Is the expected value a function $f(X)\to\mathbb R$?</p>
| BruceET | 221,800 | <p>It is totally up to the author. Some authors use a different
typeface for the $E$ and they tend to be the ones who avoid
parentheses or brackets: $\mathbf{E}X,$ $\mathbb{E}X,$ $\mathsf{E}X$, and so on (including a script E, which I've forgotten how to make).</p>
<p>It is also common to use $\mu$ when there is only ... |
2,637,337 | <p>$ABC$ is a triangle and $A_1, B_1, C_1$ are points on $BC, CA, AB$ such that $$\frac{BA_1}{A_1C}=\frac{CB_1}{B_1A}=\frac{AC_1}{C_1B}=\lambda$$</p>
<p>If $A_2, B_2, C_2$ are points on $B_1C_1, C_1A_1$, and $A_1B_1$ such that $$\frac{B_1A_2}{A_2C_1}=\frac{C_1B_2}{B_2A_1}=\frac{A_1C_2}{C_2B_1}=\frac{1}{\lambda}$$</p>
... | dezdichado | 152,744 | <p>A more thorough hint: </p>
<p>Use the theorem of sines on triangles $BB_2C_1$ and $BB_2A_1$, to conclude that:
$$\dfrac{\sin\angle B_2BA}{\sin\angle B_2BC} = \dfrac{A_1C}{BC_1}.$$</p>
<p>Then the sine version of Ceva's theorem will let you conclude that $AA_2, BB_2, CC_2$ are concurrent. This will greatly help you... |
2,637,337 | <p>$ABC$ is a triangle and $A_1, B_1, C_1$ are points on $BC, CA, AB$ such that $$\frac{BA_1}{A_1C}=\frac{CB_1}{B_1A}=\frac{AC_1}{C_1B}=\lambda$$</p>
<p>If $A_2, B_2, C_2$ are points on $B_1C_1, C_1A_1$, and $A_1B_1$ such that $$\frac{B_1A_2}{A_2C_1}=\frac{C_1B_2}{B_2A_1}=\frac{A_1C_2}{C_2B_1}=\frac{1}{\lambda}$$</p>
... | Michael Rozenberg | 190,319 | <p>$$\vec{A_2B_2}=\vec{A_2C_1}+\vec{C_1B_2}=\frac{1}{1+\frac{1}{\lambda}}\vec{B_1C_1}+\frac{\frac{1}{\lambda}}{1+\frac{1}{\lambda}}\vec{C_1A_2}=$$
$$=\frac{\lambda}{1+\lambda}\vec{B_1C_1}+\frac{1}{1+\lambda}\vec{C_1A_2}=\frac{\lambda}{1+\lambda}\left(\vec{B_1A}+\vec{AC_1}\right)+\frac{1}{1+\lambda}\left(\vec{C_1B}+\vec... |
325,588 | <p>Is there an analytic function $f$ in $\mathbb{C}\backslash \{0\}$ s.t. for every $z\ne0$: $$|f(z)|\ge\frac{1}{\sqrt{|z|}}\, ?$$</p>
| Hagen von Eitzen | 39,174 | <p>If $0$ is an essential singuarity of $f$, then by the <a href="http://en.wikipedia.org/wiki/Picard_theorem" rel="nofollow">Big Picard theorem</a>, $f(z)$ leaves out at most one value in every punctured neighbourhood of $0$.
By assumption, $f(0)\ne 0$ for all $z$, hence we find $z$ with $0<|z|<1$ and $f(z)=1$, ... |
1,992,256 | <p>I have to prove</p>
<p>$\sqrt{1} + \sqrt{2} +...+\sqrt{n} \le \frac{2}{3}*(n+1)\sqrt{n+1}$</p>
<p>by using math induction. </p>
<p>First step is to prove that it works for n = 1 , which is true.
Next step is to prove it for n + 1. We can rewrite the formula using</p>
<p>$\sum_{i=1}^{n+1} \sqrt{i}= \sum_{i=1}^{n... | Martin Sleziak | 8,297 | <p>To finish your proof by induction we could use the fact the following inequalities are equivalent to each other
\begin{align*}
\frac23(n+1)^{3/2}+\sqrt{n+1} &\le \frac23(n+2)^{3/2}\\
\frac32\sqrt{n+1} &\le (n+2)^{3/2} - (n+1)^{3/2}\\
\frac32\sqrt{n+1} &\le (\sqrt{n+2} - \sqrt{n+1}) (n+2 + \sqrt{(n+2)(n+1... |
252,820 | <p><code>Sound[]</code> generates a visual representation of notes. I would like to extract that image, only notes, without controls and borders. How can I do it?</p>
<p>Take this example</p>
<pre><code>Sound[SoundNote @@@ Transpose @ {
{"E5", "D5", "F#4", "G#4", "C#5"... | LouisB | 22,158 | <p>This example returns a <code>SparseArray</code>. Use <code>Normal</code> to make it a nested list, if you prefer.</p>
<pre><code>mat = Block[{b, n = 6},
b = Riffle[ConstantArray[1, n/2], 0];
SparseArray[{Band[{1, 2}] -> b, Band[{2, 1}] -> -b}, {n, n}]
];
MatrixForm @ mat
</code></pre>
<p><span class=... |
2,804,495 | <p>I was asked to solve this double integral:
Compute the area between $y=2x^2$ and $y=x^2$ and the hyperbolae $xy=1$ and $xy=2$ in </p>
<p>$$ \iint dx \,dy$$</p>
<p>I tried to solve it starting with considering that </p>
<p>$$x^2 \leq y \leq 2x^2 $$</p>
<p>suitabile for integration interval in $y$, obtaining the i... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Let consider the change of variables</p>
<ul>
<li>$u=x^2\implies 1\le u\le 2$</li>
<li>$v=xy\implies 1\le v\le 2$</li>
</ul>
<p>and</p>
<p>$$dudv=|J|dxdy=\begin{vmatrix}2x&0\\y&x\end{vmatrix}dxdy=2x^2dxdy\implies dxdy=\frac1{2u}dudv$$</p>
|
2,995,327 | <p>Suppose a,b ∈ Z. If 4 | <span class="math-container">$(a^2 + b^2)$</span> then a and b are not both odd.</p>
<p>So, assuming that 4 | <span class="math-container">$(a^2 + b^2)$</span> and <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are odd</p>
<p>this gives <span class="mat... | Nodt Greenish | 614,664 | <p>If you multiply correctly, the RHS leaves a rest of 2 modulo 4. So your idea was correct, you just need to recalculate.</p>
|
1,199,304 | <p>Let $M\neq \{0\}$ be a semi-simple left $R$ module .Prove that it contains a simple sub-module.</p>
<p>An $R-$ module $M$ is said to be semi-simple if every submodule of $M$ is a direct summand of M
<strong>My solution</strong></p>
<p>Since $M\neq \{0\}$; $\exists m\in M$ such that $m\neq 0$.Then I can consider th... | egreg | 62,967 | <p>First show that every submodule of a semisimple module is also semisimple.</p>
<p>Let $M$ be semisimple, $N$ a submodule of $M$ and $L$ a submodule of $N$. Since $M$ is semisimple, we have $N\oplus N'=M$ and also $L\oplus L'=M$. Consider then $L''=L'\cap N$. Then $L\cap N=L\cap L'\cap N=\{0\}$. If $x\in N$, we have... |
2,545,516 | <p>So I have to assess the convergence of $$\displaystyle\sum_{n=1}^{\infty}\sin\left(\displaystyle\frac{1}{\sqrt{n}}\right).$$</p>
<p>I'm told that it diverges, but can't really see why.</p>
<p>The divergence test doesn't really help, because
$\lim\limits_{x\to\infty}\displaystyle\frac{1}{\sqrt{n}}=0$, so</p>
<p>... | qualcuno | 362,866 | <p>You can do a limit comparison with <span class="math-container">$\displaystyle\sum\limits_{n\geq1}\dfrac{1}{\sqrt{n}}$</span>, since</p>
<p><span class="math-container">$$
\lim_{n\to\infty}\sqrt{n}\sin\left(\frac{1}{\sqrt{n}}\right) = 1
$$</span></p>
|
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Mozibur Ullah | 35,706 | <p>Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, <em>Geometry and the Imagination</em>.</p>
<blockquote>
<ol>
<li>The simplest curve is the the straight line and its curvature obviously should be zero.</li>
</ol>
</blockquote... |
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Nitin Nitsure | 148,928 | <p>Curvature can be very easily pictured using `geodesic quadrilateral gaps',
which can be more generally used to recover the torsion tensor, and if the torsion is
identically zero, then the curvature tensor, for a manifold equipped with an affine connection.</p>
<p>In the special case of an oriented Riemannian surface... |
275,539 | <p>Kind of leading on from my other question, how would I solve for $i$? Or how would I check that it is possible to have such an $i$?</p>
<p>First I had to check for all $2^i$ and clearly this doesn't happen as all $2^i$ are even and so I will just get even $x's$ such that $2^i \equiv x \mod 28$. So the next one I go... | lab bhattacharjee | 33,337 | <p><a href="http://mathworld.wolfram.com/CarmichaelFunction.html" rel="nofollow">Carmichael Function</a> is more useful than <a href="http://mathworld.wolfram.com/TotientFunction.html" rel="nofollow">Totient Function</a>, while dealing with composite numbers like $28$.</p>
<p>For
$\phi(28)=\phi(7)\cdot\phi(4)=6\cdot2... |
1,368,988 | <p>I was thinking about different ways of finding $\pi$ and stumbled upon what I'm sure is a very old method: dividing a circle of radius $r$ up into $n$ isosceles triangles each with radial side length $r$ and central angle $$\theta=\frac{360^\circ}{n}$$ Use $s$ for the side opposite to $\theta$.</p>
<p><img src="htt... | Bhaskara-III | 246,676 | <blockquote>
<p>Just recall double angle trig. identity: $\cos 2x=1-2\sin^2x$,</p>
</blockquote>
<p>Starting from,
$$\begin{align}
\lim_{n\to \infty}\frac{n}{2}\sqrt{2-2\cos\left(2\pi/n\right)}\\=\lim_{n\to \infty}\frac{n}{2}\sqrt{2-2+4\sin^2\left(\pi/n\right)}\\
=\lim_{n\to \infty}\frac{n}{2}\cdot 2\sin\left(\pi/n\... |
103,675 | <p>I have defined a recursive sequence</p>
<pre><code>a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
</code></pre>
<p>because I want to calculate the <code>Limit</code> for this sequence when n tends towards infinity.</p>
<p>Unfortunately I get a <code>recursion exceeded</code> error when doing:</p>
<pre><code>Limit[a[n... | bbgodfrey | 1,063 | <p>Alternatively, and perhaps more directly, use</p>
<pre><code>RSolve[{a[n] == Sqrt[3] + 1/2 a[n - 1], a[0] == 1}, a[n], n]
(* {{a[n] -> 2^-n (1 - 2 Sqrt[3] + 2^(1 + n) Sqrt[3])}} *)
Limit[a[n] /. %[[1]], n -> Infinity]
(* 2 Sqrt[3] *)
</code></pre>
|
1,903,473 | <p>Given three permutations $p_1,p_2,p_3$ of $\{1,2,\ldots,n^3+1\}$, prove that two of them have a common subsequence of length $n+1$.</p>
<p>I have tried to solve this using the pigenhole principle but I didnt progress too much, any help would be appreciated</p>
<p>edit: when I say subsequence I mean that there are ... | openspace | 243,510 | <p>That's not true.
Consider three permutations :
$$p_1 = (1, \dots, n^3+ 1)$$
$$p_2 = (n^3+1, \dots, 1)$$
$$p_3 = (1, n^3, 3, n^3-2, \dots)$$
where $p_3[i] = p_{i \bmod 2}[i]$</p>
|
1,903,473 | <p>Given three permutations $p_1,p_2,p_3$ of $\{1,2,\ldots,n^3+1\}$, prove that two of them have a common subsequence of length $n+1$.</p>
<p>I have tried to solve this using the pigenhole principle but I didnt progress too much, any help would be appreciated</p>
<p>edit: when I say subsequence I mean that there are ... | arghbleargh | 362,140 | <p>You can prove this with ideas that are used in proving the Erdos-Szekeres theorem. Without loss of generality, we can assume $p_1$ is the identity permutation (otherwise just relabel numbers). Then, if either of $p_2$ or $p_3$ has an increasing subsequence of length $n + 1$, we are done. So assume their longest incr... |
3,497,420 | <p>Consider the function <span class="math-container">$$f(x,y)=x^6-2x^2y-x^4y+2y^2.$$</span> The point <span class="math-container">$(0,0)$</span> is a critical point. Observe,
<span class="math-container">\begin{align*}
f_x & = 6x^5-4xy-4x^3y, f_x(0,0)=0\\
f_y & = 2x^2-x^4+4y. f_y(0,0)=0\\
f_{xx} & = 30x... | bjorn93 | 570,684 | <p>You have <span class="math-container">$f(0,0)=0$</span>. You can find both positive and negative values of <span class="math-container">$f(x,y)$</span> in any region around <span class="math-container">$(0,0)$</span>, which means that you don't have a local extremum at this point:
<span class="math-container">$$f(x,... |
2,235,427 | <p>I am trying to characterize the continuum $\mathcal{C}$ using only the notion of midpoint, i.e. the operation $\mu : \mathcal{C}\times\mathcal{C} \to \mathcal{C}$ assigning to each pair of points the midpoint between them. I thought of defining “unbounded interval” as a set $I \subseteq \mathcal{C}$ such that $I$ as... | Misha Lavrov | 383,078 | <p>Working with the usual definition of $\mathbb R$ (and assuming choice), we can define a set $I$ that's not an unbounded interval, but for which both $I$ and $\mathbb R \setminus I$ are closed under taking midpoints. (I use $\mathbb R$ to refer to the usual notion of realnumberness to distinguish from $\mathcal C$ wh... |
2,235,427 | <p>I am trying to characterize the continuum $\mathcal{C}$ using only the notion of midpoint, i.e. the operation $\mu : \mathcal{C}\times\mathcal{C} \to \mathcal{C}$ assigning to each pair of points the midpoint between them. I thought of defining “unbounded interval” as a set $I \subseteq \mathcal{C}$ such that $I$ as... | William Balthes | 231,063 | <p>Are you putting metric structure on these midpoints; such as as a weakened version of midpoint convexity.</p>
<p>Or just that the midpoint functions values exist in that appropriates, F(c/2+c1/2) exists in the structure somewhere between the c1 and c2 c1
|
1,572,045 | <p>This is maybe a stupid question, but I want to find the roots of:</p>
<blockquote>
<p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$</p>
</blockquote>
<p>What that I did:</p>
<p>$$\underbrace{2(x+2)(x-1)(x-1)(x-1)}_{A}-\underbrace{3(x-1)(x-1)(x+2)(x+2)}_{B}=0$$</p>
<p>So the roots are when $A$ and $B$ are both zeros when... | user236182 | 236,182 | <p>Use the Distributive Property.</p>
<p>$$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2$$</p>
<p>$$=\underbrace{(x+2)(x-1)^2}_{\text{common factor}}\left(2(x-1)\right)-\underbrace{(x+2)(x-1)^2}_{\text{common factor}}(3(x+2))$$</p>
<p>$$=(x+2)(x-1)^2\left(2(x-1)-3(x+2)\right)$$
$$=(x+2)(x-1)^2(-(x+8))$$</p>
|
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| mounir ben salem | 244,326 | <p>you can use: $$ \cos(x)= \sqrt{1-\sin(x)^{2}}$$ $$ \text{then let}\qquad \sin(x)=t\\
\sqrt{1-t^{2}}+t-1=0$$ now you can continue</p>
|
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Bernard | 202,857 | <p>$\cos x+\sin x=\sqrt2\cos\Bigl(x-\dfrac\pi4\Bigr)$, hence the equation is equivalent to:
$$\cos\Bigl(x-\frac\pi4\Bigr)=\frac1{\sqrt2}\iff x-\frac\pi4\equiv\pm\frac\pi4\mod 2\pi\iff x\equiv 0,\,\frac\pi2\mod2\pi.$$</p>
|
713,521 | <p>There are so many notations for differentiation. Some of them are:
$$
f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad
D f(x)\qquad y^\prime\qquad D_x f(x)
$$
Why are there so many ways to say "the derivative of $f(x)$"? Is there a specific use for each notation? What is the differ... | Slade | 33,433 | <p>For the most part, the things you've written are equivalent, and the reason there are so many is partly historical, partly practical (e.g. $D_x$ is better notation when one is using the language of operators or partial derivatives, $y'$ saves space when it's unambiguous, etc.).</p>
<p>But there are two really huge ... |
846,108 | <p>How do you solve this equation: $2x+8=6x-12$ by using the guess and check method?</p>
<p>I divide $2x+8$ and I get $4$ then I divide $6x-12$ and I get $-2$ but I don't know what to do next or is it wrong?</p>
| Joel | 85,072 | <p>Your instructor wants you to try plugging in a bunch of numbers to guess the correct value. Just by eyeballing this, we might try 4 or 5. If you plug in $5$ on the left you get $$2(5)+8=18$$ and on the right you get $$6(5)-12 = 30 - 12 = 18.$$ This tells us that $x=5$ is a solution to this equation.</p>
<p>Remember... |
2,353,142 | <p>Solve:
$$(\cot^{-1} (x))^2 - 4\cot^{-1} (x) + 3 \geq 0$$.</p>
<p>My Attempt:
$$(\cot^{-1} (x))^2 - 4\cot^{-1} (x) + 3 \geq 0$$.
Let $\cot^{-1} (x)=t$. then</p>
<p>$$t^2-4t+3\geq 0$$
$$(t-3)(t-1)\geq 0$$
Either, $\cot^{-1} (x) \leq 1$</p>
<p>Or, $\cot^{-1} (x) \geq 3$</p>
<p>I solved till here, but couldn't get ... | Michael Rozenberg | 190,319 | <p>It's true for $p=2$ only because for real $p$ we need $x>0$ by definition of $x^p$ and our inequality can not hold for all reals $x$ and $y$.</p>
<p>If $p>2$ and odd then $x=y=-1$ gets a contradiction.</p>
<p>If $p>2$ and even then $x=y\rightarrow0^+$ gets a contradiction again. </p>
|
691,494 | <p>Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix </p>
<p>$\mathbf{P}= \begin{bmatrix}
\frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em]
\frac{2}{3} & 0 & \frac{1}{3} & 0\\[0.3em]
\frac{2}{3} & 0 & 0 & \frac{1}{3}\\[0.3em]
... | Rodrigo Ribeiro | 44,681 | <p>Usually, my approach to this kind of question is to solve a very simple recurrence.</p>
<p>Just look to $ E_{0} N(3)$ but conditioned in each of the two possible first steps to get (I will write $N$ instead of $N(3)$</p>
<p>$$E_0 N = \frac{2}{3}E_0N + \frac{1}{3}E_1N +1$$</p>
<p>The term +1 shows up because once ... |
1,066,484 | <p>Given a simple connected bipartite graph $G$ with degree of vertices equal to $k$, where $k\ge 2$. Prove that there is no cut vertex exist in $G$. </p>
<p>Cut vertex $v$ here is a vertex which make the graph induced have number of connected component $>1$ when $v$ is removed.</p>
<p>I have tried to prove by con... | Leen Droogendijk | 95,972 | <p>Hint:</p>
<p>Show that the number of edges in the original graph is a multiple of $k$.</p>
<p>Show that the removal of one vertex removes $k$ edges.</p>
<p>Show that the number of edges in each component (after the removal) still is a multiple of $k$.</p>
<p>Conclude that the original graph must already have bee... |
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | Clement C. | 75,808 | <p>Taylor, first order, would do. $\arctan u = u+o(u)$, $\ln(1+u) = u+o(u)$, so than $\ln(1+\arctan u) = u+o(u)$ when $u\to 0$. So your limit is $e^{\frac{1}{u}\ln(1+\arctan u)} = e^{1+o(1)}$, with $u=\frac{x}{2}$; $$e^{\frac{1}{u}\ln(1+\arctan u)} \xrightarrow[u\to0]{} e^1 = e.$$</p>
<hr>
<p><em>Remark: there may be... |
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | Paramanand Singh | 72,031 | <p>First and foremost you can get rid of the $x/2$ (it is a pain to type fractions) by putting $t = x/2$ and using the fact that as $x \to 0$ the variable $t \to 0$. We thus need to evaluate $$L = \lim_{t \to 0}(1 + \arctan t)^{1/t}\tag{1}$$ We can now proceed in your manner
\begin{align}
\log L &= \log\left\{\lim_... |
872,889 | <p>What determinant is zero? What equation does this give for the plane?</p>
<p>I need some help here, am pretty stuck</p>
| Steven Stadnicki | 785 | <p>Expanding my comments into an answer: by distributing the divisions by $a_0$, $a_1$, $\ldots$ successively you can rewrite such an upward continued fraction in the equivalent form $\frac1{a_0}+\frac1{a_0a_1}+\frac1{a_0a_1a_2}+\cdots$. This is known as the <em><a href="http://en.wikipedia.org/wiki/Engel_expansion" r... |
2,024,997 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$</p>
</blockquote>
<p>I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?</p>
| Caleb Stanford | 68,107 | <p>Since the limit of both the top and bottom is $\infty$ alone, l'hopital's rule gives us
\begin{align*}
\lim_{x \to \infty}\frac{\log_{1.1} x}{x}
&= \lim_{x \to \infty}\frac{\ln x}{(\ln 1.1) x} \\
&= \lim_{x \to \infty}\frac{(1/x)}{(\ln 1.1)} \\
&= 0.
\end{align*}</p>
|
2,024,997 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$</p>
</blockquote>
<p>I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?</p>
| Bernard | 202,857 | <p>Use the definition: $\;\log_{1.1}x=\dfrac{\ln x}{\ln 1.1}$, so
$$\frac{\log_{1.1}x}{x}=\frac 1{\ln 1.1}\dfrac{\ln x}{x}\xrightarrow[x\to+\infty]{}\frac 1{\ln 1.1}\cdot 0=0.$$</p>
|
1,919,912 | <blockquote>
<p>Let $D$ be the Integral Domain with characteristic $m>0$. Prove that $m$ is prime.</p>
</blockquote>
<p>My Proof: </p>
<p>Since the characteristic of $D$ is $m$, $m\cdot b=0$ for all $b\in D$ and if $n\cdot b=0$ for all $b\in D$, then $m\leq n$. </p>
<p>Assume that $m$ is composite number. Then ... | Exit path | 161,569 | <p>To simplify things, let $m$ be the characteristic of $D$. Let $m=nk$ be a factorization of $m$. Then $m\cdot 1=(nk)\cdot 1=(n \cdot 1)(k\cdot 1)=0$. Since $D$ is an integral domain, either $n \cdot 1=0$ or $k \cdot 1=0$. But $m$ is the least integer for which $m \cdot 1=0$, implying $n=1$ or $k=1$, so $m$ is prime. ... |
2,068,951 | <p>I'm interested in proving the following claim:</p>
<p>There exists a sequence of natural numbers $\left(a_{n}\right)_{n=1}^{\infty}$ such that
$$\lim_{n\to\infty}\left(1-\frac{1}{2^{n}}\right)^{a_{n}}=\frac{1}{2}
$$</p>
<p>I've studied a fair amount of calculus and algebra, yet I've never encountered such a probl... | Barry Cipra | 86,747 | <p>Try </p>
<p>$$a_n=\lfloor2^n\ln2\rfloor$$</p>
<p>and use the inequalities $2^n\ln2-1\lt a_n\le2^n\ln2$ in the Squeeze Theorem: Since $1-{1\over2^n}\lt1$, we have</p>
<p>$$\left(1-{1\over2^n}\right)^{2^n\ln2}\le\left(1-{1\over2^n}\right)^{a_n}\lt\left(1-{1\over2^n}\right)^{2^n\ln2-1}$$</p>
<p>The left- and right... |
671,407 | <p>I have problem with equation: $4^x-3^x=1$. </p>
<p>So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other solutions? </p>
| Blaubaer | 126,764 | <p>Show that $4^x$ grows much faster than $3^x$.</p>
|
3,057,198 | <p><span class="math-container">$$\iint_{G}\!x^2\,\mathrm{d}x\mathrm{d}y$$</span></p>
<p>where <span class="math-container">$G := \left\{(x,y)\in\mathbb{R}^{2}\,;\,|x|+|y| \le 1\right\}$</span></p>
<p>How does one go about finding the boundaries of these types of integrals? I did fail at searching for examples like t... | Rafa Budría | 362,604 | <p>You have to solve the inequalities for the four cases:</p>
<p><span class="math-container">$x\geq0\,\land\,y\geq0\,\land x+y\leq1$</span>, the triangle with vertices at <span class="math-container">$(0,0),(0,1),(1,0)$</span>, limited by the function <span class="math-container">$y=1-x$</span></p>
<p><span class="m... |
3,057,198 | <p><span class="math-container">$$\iint_{G}\!x^2\,\mathrm{d}x\mathrm{d}y$$</span></p>
<p>where <span class="math-container">$G := \left\{(x,y)\in\mathbb{R}^{2}\,;\,|x|+|y| \le 1\right\}$</span></p>
<p>How does one go about finding the boundaries of these types of integrals? I did fail at searching for examples like t... | StubbornAtom | 321,264 | <p>Sketch the region <span class="math-container">$G$</span>. </p>
<p><img src="https://i.stack.imgur.com/Z8wxO.jpg" alt="enter image description here"></p>
<p>It is now clear that <span class="math-container">$$G=\left\{(x,y)\in\mathbb R^2: |y|\le 1-|x|\,,\,|x|\le 1\right\}$$</span></p>
<p>So, </p>
<p><span class=... |
3,031,290 | <p>Can you choose <span class="math-container">$11$</span> different numbers among them so that the numbers <span class="math-container">$|a_1-a_2|, |a_2-a_3|,\ldots,|a_{10}-a_{11}|,|a_{11}-a_{1}|$</span> are all different. The smartest thing that my dumbest mind could accomplish is that all those differences are <span... | Jam | 161,490 | <p><strong>@Saulspatz</strong> has shown an example but I've found a methodical approach to finding one that could be done by hand in a few minutes to give you an answer in a competition. It could also be easily computerised.</p>
<p>Start by laying out a grid with columns, <span class="math-container">$D=1,2,\ldots11$... |
4,651,596 | <p>I know the proof of the "<a href="https://en.wikipedia.org/wiki/Doubling_the_cube" rel="nofollow noreferrer">Doubling the cube problem</a>". What is used there is the fact that if a number <span class="math-container">$a$</span> is constructible then <span class="math-container">$[\mathbb{Q}(a):\mathbb{Q}]... | Michael Weiss | 79,741 | <p>Cox's <em>Galois Theory</em>, Example 10.1.13 (p.263) gives the example <span class="math-container">$f(x)=x^4-4x^2+x+1$</span>. It is irreducible, so if <span class="math-container">$f(a)=0$</span> then <span class="math-container">$[\mathbb{Q}(a):\mathbb{Q}]=4$</span>. However, the splitting field of <span class="... |
4,177,639 | <p>I have an object with known coordinates in in 3D but on the ground (<code>z=0</code>). The object has a direction vector. My goal is to move this object on the ground (so <code>z</code> stays <code>0</code>) using its direction vector and via randomly-generated velocity vectors with one condition: I want to ensure t... | IV_ | 292,527 | <p><span class="math-container">$$a^x+b^x=1$$</span></p>
<p>Substitute <span class="math-container">$x=\frac{\ln(t)}{\ln(a)}$</span>:</p>
<p><span class="math-container">$$a^{\frac{\ln(t)}{\ln(a)}}+b^{\frac{\ln(t)}{\ln(a)}}=1$$</span></p>
<p><span class="math-container">$$t+t^{\frac{\ln(b)}{\ln(a)}}=1$$</span></p>
<p>S... |
87,437 | <p>Let <span class="math-container">$R$</span> be a rectangular region of the integer lattice <span class="math-container">$\mathbb{Z}^2$</span>,
each of whose unit squares is labeled with a number
in <span class="math-container">$\lbrace 1, 2, 3, 4, 5, 6 \rbrace$</span>.
Say that such a labeled <span class="math-conta... | Marzio De Biasi | 35,419 | <p>While browsing the recreational-mathematics tag I noticed this old question again.</p>
<p>In 2012 I published on my blog a draft paper in which I give another example of a (bigger) board with two distinct Hamiltonian cycles (but domotorp was quicker to answer).</p>
<p>In the paper I also prove that the <em>Rolling... |
3,691,255 | <p>Pierre runs a game at a fair, where each player is guaranteed to win $10. </p>
<p>Players pay a certain amount each time they roll an unbiased die, and must keep rolling until a ‘6’ occurs. </p>
<p>When a ‘6’ occurs, Pierre gives the player $10 and the game concludes. </p>
<p>On average, Pierre wishes to make a p... | gnasher729 | 137,175 | <p>In the long term, one out of six throws gives a six, costing Pierre 10 dollars and ending a game. Pierre wants to make 2 dollars profit per game, so he must ask for 12 dollars for each six thrown, so he charges 2 dollars per throw. No sums needed. </p>
|
1,908,923 | <p>Let $X$ be a Riemannian manifold*, and $S$ a compact submanifold of $X$. </p>
<p>Assume there exists an <strong>open, dense</strong> subset $Y$ of $\,X$, such that for any element $y \in Y$, there exists a unique element in $S$ closest to $y$; i.e there is a function $\tilde s:Y \to S$ such that $$ d(y,\tilde s(y))... | Asaf Shachar | 104,576 | <p>I am adding some details to the answer of HK Lee:</p>
<p>We assume $X$ is geodesically convex, i.e there is a minimizing geodesic between every two points. (In particular, our argument holds for any complete manifold).</p>
<p>Let $x \in X$, and suppose $s_0$ is a closest point to $x$ in $S$. Let $c:[0,1] \to X$ be... |
1,257,598 | <p>Suppose A is a family of subsets of R with the property that the intersection of any two sets in A is finite. Show that $|A|\leq 2^{\aleph_0}$.</p>
<p>I was told that choosing a countable $D \subset B$ for all $B \in A$ would be helpful. I'm just really not sure where to go with this. Any hints would be appreciated... | Camilo Arosemena-Serrato | 33,495 | <p><strong>Hint:</strong> Prove that the set $[\Bbb R]^{<\omega}$, the set of all finite subsets of $\Bbb R$, has size $2^{\aleph_0}$.
Fix $C_0\in A$, then prove that for each $C\in [\Bbb R]^{<\omega}$, the set $A_C:=\{B\in A:B\cap C_0=C \}$ has size $\leq2^{\aleph_0}$. Notice that $\{A_C:C\in [\Bbb R]^{<\om... |
1,297,690 | <p>If I have a program that creates, let's say, one billion integers, with each having a pure $50 - 50$ chance to be one or zero,</p>
<p>what is the chance of finding $x$ zeros in a row?</p>
<p>for brownie points, instead of the program creating a set billion numbers, what would the equation be with $z$ numbers?</p>
| alkabary | 96,332 | <p>consider with just 10 integers $$\large{x_1x_2x_3x_4x_5x_6x_7x_8x_9x_{10}}$$ where all $\large{x_i}$ is either $0$ or $1$.</p>
<p>What is the chance of finding $\large{x_1x_2x_3x_4}$ all zeros ?</p>
<p>It is just $\large{(\frac{1}{2})^4 = \frac{1}{16}}$. Now you just apply this to your question</p>
|
222,596 | <p>I would like to find a temperature by knowing the enthalpy, is this possible?
This is what i've tried so far:</p>
<pre><code>V1 = 150;
V2 = 4;
T1 = 15 + 273;
Enthalpy
h[T_] := QuantityMagnitude[
ThermodynamicData["Air",
"Enthalpy", {"Temperature" -> Quantity[T, "Kelvins"]}]]
h1 = h[T1]
sol = ... | flinty | 72,682 | <p><em>Mathematica</em> is missing enthalpy data below 60K. Also evaluating the <code>ThermodynamicData</code> inside the <code>Solve</code> is slow. Try this:</p>
<pre><code>enthalpy[t_?NumericQ] :=
QuantityMagnitude[
QuantityMagnitude[
ThermodynamicData["Air",
"Enthalpy", {"Temperature" -> Quantity[t,... |
4,177,829 | <p>Given angles <span class="math-container">$0<\theta_{ij}<\pi$</span> for <span class="math-container">$1\leq i<j\leq k$</span>, what conditions are there on the angles to ensure that there exists <span class="math-container">$k$</span> unit vector <span class="math-container">$v_i\in \mathbb R^k$</span> so ... | NN2 | 195,378 | <p>Let us represent the vectors in question as directed line segments <span class="math-container">$\{\overrightarrow{OA}_i\}_{1 \le i \le k} $</span> from the origin <span class="math-container">$O$</span> to the points <span class="math-container">$A_i$</span> on the unit sphere <span class="math-container">$\mathcal... |
95,964 | <p>On the page 43 of <em>Real Analysis</em> by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties:</p>
<ol>
<li>$m(E)$ is defined for each subset $E$ of real numbers.</li>
<li>For an interval $I$, $m(I) = l(I)$ (the length of $I$).</li>
<li>If $\{E_... | Jonas Meyer | 1,424 | <p>I refer you to the MathOverflow question "<a href="https://mathoverflow.net/questions/45784/does-pointwise-convergence-imply-uniform-convergence-on-a-large-subset">Does pointwise convergence imply uniform convergence on a large subset?</a>" for proofs and references for the following:</p>
<blockquote>
<p>Assuming... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Blue | 409 | <p>A couple of my previous answers to similar questions:</p>
<p>Find a rubber chicken(!) and introduce the kids to $\pi$:
<a href="https://math.stackexchange.com/a/395933/409">https://math.stackexchange.com/a/395933/409</a> (It won't take a half-hour, but it's a good way to warm-up the crowd. :)</p>
<p>This exercise ... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| KCd | 619 | <ol>
<li><p>Discuss the birthday paradox if there are at least 23 people in the room. In fact, ask the teacher in advance if he/she knows from student records if two students share the same birthday (an illustration with the students in the room won't go over well if you try it and nobody shares a birthday).</p></li>
... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Loki Clock | 58,287 | <p>Describe multiplication on a 12-hour clock, pointing out that 3*4=12 makes it unlike regular multiplication. Ask them which clocks don't have two numbers that can be multiplied to get the number of hours on the clock.</p>
<p>Taking (topological) dual polyhedra by letting new points be face centers and connecting ce... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Eric Jablow | 70,913 | <ol>
<li>Make a bagatelle and drop balls down it. Use this to demonstrate the central limit theorem, suitably simplified to "Large amounts of random anything look like a bell curve." Then, measure your students' heights and compare.</li>
<li>Try logic puzzles and games.
<ol>
<li>Nim</li>
<li>The Monty Hall Problem</l... |
366,311 | <blockquote>
<p>Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.</p>
</blockquote>
<p>I'm not familiar with proving divergent sequence. Do anyone have any des? Thank you.</p>
| Community | -1 | <p>For $\epsilon=\frac{1}{2}$ and $\forall n , \displaystyle|x_{2n}-x_n|=\sum_{k=n+1}^{2n}\frac{1}{k}\geq n\times\frac{1}{2n}=\epsilon$ hence the sequence $(x_n)$ is divergent since it's not a Cauchy sequence.</p>
|
1,005,193 | <p>My problem is as follows:</p>
<p>I have a point $A$ and a circle with center $B$ and radius $R$. Points $A$ and $B$ are fixed, also $A$ is outside of the circle. A random point $C$ is picked with uniform distribution in the area of disk $B$. My question is how to calculate the expected value of $AC^{-4}$. I am work... | Did | 6,179 | <blockquote>
<p>I don't know what limits to use.</p>
</blockquote>
<p>Note that $x=w/u$, $y=u$, $z=\sqrt{v}$ with $0\leqslant x,y,z\leqslant1$ hence the domain of integration is $$0\leqslant w/u,u,\sqrt{v}\leqslant1,$$ or, equivalently, $$0\leqslant w\leqslant u\leqslant1,\qquad0\leqslant v\leqslant1.$$</p>
<blockq... |
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| Pietro Majer | 6,101 | <p>Well, understanding mathematics has different levels: Understanding is, mainly, pretending to understand. If you are able to cheat your professors, then your students, and your colleagues, it's OK. If you succeed to cheat yourself, then it means you went a bit too far. </p>
|
188,087 | <p>Is there a function that can extract a list of variables in an expression?
For example, assume we have an expression</p>
<pre><code>x^2+y^3+z
</code></pre>
<p>This expression has variables x, y and z. The result should be</p>
<pre><code>{x, y, z}
</code></pre>
<p>. Is there a way to get this?</p>
| Xminer | 61,541 | <p>I like the following approach x):</p>
<pre><code>expr = x^2 + y^3 + z;
Select[DeleteDuplicates@Level[expr, Depth@expr], Head[#] == Symbol &]
</code></pre>
<p>the result is:</p>
<pre><code>{x, y, z}
</code></pre>
|
1,319,476 | <p>This is a question related to another posted question:</p>
<p>The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: </p>
<p>"Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$,</p>
<p>so: $ \cos x+i\sin x=0+1⋅i$</p>
<p>compare real and imaginary parts
$\sin(x)=1$
and
$\cos(x)=0$</p>
<p... | Deepak | 151,732 | <p>I always find it easier to use a fixed method, and I thought you might find this explanation easier, so I'm posting it. </p>
<p>Start by putting everything into exponential form. Now $i = e^{\frac{i\pi}{2}}$. You can derive this from $e^{i\pi} = -1$ and taking square roots on both sides.</p>
<p>Now note that for <... |
1,913,320 | <blockquote>
<p>Let <span class="math-container">$A=(a_{ij})_{n\times n}$</span> and <span class="math-container">$A=(a_{ij})_{n\times n}$</span> be two upper triangular matrices, i.e. <span class="math-container">$a_{ij}=b_{ij}=0$</span> whenever <span class="math-container">$i>j$</span>.</p>
<p><span class="math-c... | DonAntonio | 31,254 | <p>$$2\cdot3\cdot5\cdot7\cdot11\cdot13+1=59\cdot509$$</p>
|
2,745,918 | <p>The numbers $1,2, \ldots, n$ are written in a board, with $n \in \mathbb{N}$. In every move, we can choose two numbers of the board, find their $\rm lcm$, and replace the two numbers with it. After $k$ moves, we find the sum of the numbers in the board, and we name it $S$. Find the minimum and the maximum value of $... | Joffan | 206,402 | <p>Some hints (all numbers are $\in \Bbb N$).</p>
<ul>
<li>for any $k$, ${\rm lcm}(k,1)=k$.</li>
<li>for any $j,k,$ ${\rm lcm}(j,jk)=jk$.</li>
<li>the $\rm lcm$ of a prime number $p$ and any other distinct number $k$ is never less than the prime number. If $k>1,$ ${\rm lcm}(p,k)>p$.</li>
<li>if we have a set of ... |
1,005,154 | <p>I don't know how to advance in the following <em><strong>problem</strong></em>:</p>
<p>Let $X$, $Y$ and $Z$ independent random variables equally distributed with uniform distribution over $[0,1]$.</p>
<ul>
<li>Find the joint pdf of $W:=XY$ and $V:=Z^2$.</li>
</ul>
<hr>
<p><em><strong>I tried to</strong></em> ans... | Vladimir Vargas | 187,578 | <p>Notice that:</p>
<p>$$f_{WVU}(w,v,u)=|\boldsymbol{J(h)}|f_{XYZ}(h(x,t,z))=|\boldsymbol{J(h)}|f_X\left(\dfrac{w}{u}\right)\chi_{[0,1]}(w)f_Y(u)\chi_{[w,1]}(u)f_Z(\sqrt{v})\chi_{[0,1]}(v).$$</p>
|
3,145,896 | <h1>Solve for <span class="math-container">$x$</span></h1>
<p>I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone show me how to solve for <span class="math-container">$x$</span> in this... | Claude Leibovici | 82,404 | <p><em>Just for the fun of it !</em></p>
<p>Since heropup already gave the answer, let us do the same using <strong>one single</strong> iteration using <a href="http://numbers.computation.free.fr/Constants/Algorithms/newton.html" rel="nofollow noreferrer">high order methods</a> with <span class="math-container">$x_0 ... |
2,194,190 | <p>I need to check the irreducibility of $p(x) \in F[x]$, where $F$ is a finite field.
I have read and checked on several exercises on the internet. Their solutions are as follows:</p>
<p>For instance, let $p(x)$ an arbitrary polynomial in $\mathbb{Z}_5[x]$. </p>
<p>If $p(x)$ has no zeros in $\mathbb{Z}_5$, then they... | Akash Patalwanshi | 168,676 | <p>Let $D$ be an integral domain with unity, a polynomial $f(x) ∈ D[x]$ such that $deg(f(x)) ≥ 1$ is irreducible polynomial in $D[x]$, if whenever $f(x) = g(x) • h(x)$ then either $deg((g(x)) = 0$ or $deg((h(x)) = 0$. </p>
<p>There is one more thing which is irreducible element in Integral domains. Let $R$ be a CRU (c... |
493,104 | <p>I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$, we can write the polar equation of the ellipse as $r = \sqrt{a^2 \cos^2{\thet... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Putting $x=r\cos\theta,y=r\sin\theta$</p>
<p>$$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1,$$</p>
<p>$$r^2=\frac{a^2b^2}{b^2\cos^2\theta+a^2\sin^2\theta}=b^2\frac{\sec^2\theta}{\frac{b^2}{a^2}+\tan^2\theta}$$</p>
|
164,152 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/33215/what-is-48293">What is 48÷2(9+3)?</a> </p>
</blockquote>
<p>Please, look at the picture?</p>
<p><img src="https://i.stack.imgur.com/3Tauh.jpg" alt="http://s16.radikal.ru/i190/1206/3b/7411739c6d... | Jorge Leitao | 25,780 | <p>The reason why calculators are giving different results is because one is calculating,</p>
<p>$$\frac{6}{2\cdot(2+1)} = 1$$</p>
<p>while the other is calculating,</p>
<p>$$\frac{6}{2}\cdot(2+1) = 9$$</p>
<p>Depending on what you want to calculate, you could obtain different answers.</p>
|
1,957,453 | <p>Can you please help me with this question?</p>
<p>Question: Find the radius of curvature, and the equation of the osculating circle, for the following curve for <span class="math-container">$t\geq0$</span>.</p>
<p><span class="math-container">$r(t) = \sin(\sqrt{e^t+1}) \hat{i} - \cos(\sqrt{e^t+1}) \hat{j} + 0 \hat... | Michael Hoppe | 93,935 | <p>It is much easier to tackle the problem in common. The curve $r$ in question is a curve in the Euclidian plane. Its curvature $\kappa_r$ is the normal component of the acceleration divided by the square of its velocity. Define $J(a,b):=(-b,a)$ to be the rotation by $\pi/2$ as usual. So
$$\kappa_r=\frac{\langle \ddo... |
3,440,093 | <p>The problem is minimize over all <span class="math-container">$\theta \in \mathbb{R}^n$</span></p>
<p><span class="math-container">$$\frac{1}{2} ||Y - \theta||^2$$</span> subject to <span class="math-container">$A \theta = 0$</span> where <span class="math-container">$A$</span> is <span class="math-container">$m \t... | pre-kidney | 34,662 | <p>Another way to phrase the question is to find which vector in the subspace <span class="math-container">$V=\ker A$</span> is closest to the vector <span class="math-container">$Y\in \mathbb R^n$</span>.</p>
<p>Suppose we can find <span class="math-container">$v\in V$</span> such that <span class="math-container">$Y... |
2,210,893 | <p>A lot of times when proving for example inequalities like $$x \leq y$$
for real numbers $x,y$ the argument looks like
$$x \leq y + \varepsilon$$
for all $\varepsilon > 0$, hence $x \leq y$. </p>
<p>Now this is obviously very intuitive, but is there a "proof" that this conclusion is correct? And is it always suf... | fleablood | 280,126 | <p>A silly direct proof (for those who don't) like proofs by contradiction).</p>
<p>$x\le y +\epsilon$ for all $\epsilon > 0$.</p>
<p>Let $\tau =y-x $.</p>
<p>$y= x +\tau $ so $\tau \le 0$. </p>
<p>So $y \le x $</p>
<p>$-\tau \ge 0$. So as $x\ge y $ we have $x+(-\tau) \ge y+(-\tau) =y-y+x = x $. So $-\tau \le... |
2,359,700 | <p>Given the vector space, $ C(-\infty,\infty)$ as the set of all continuous functions that are always continuous, is the set of all exponential functions, $U=\{a^x\mid a \ge 1 \}$, a subspace of the given vector space?</p>
<p>As far as I'm aware, proving a subspace of a given vector space only requires you to prove c... | StackTD | 159,845 | <p>If you're going over these kind of limits, you have probably heard of the <em>path test</em>.</p>
<p>If you can find two different paths leading to $(0,0)$ but where the limits along those paths are different, then the initial, two-variable limit does not exist.</p>
<p><strong>Hint</strong>: looking at the powers ... |
3,099,815 | <p>I need some help on how to approach this problem. I can't seem to find any examples that help me understand this, so if anyone has an approach example to post I would be very grateful:</p>
<p>"Consider a relation <span class="math-container">$R$</span> defined on the set of integers. Determine for the following if ... | MJD | 25,554 | <p>Let's ask if <span class="math-container">$R$</span> is symmetric. For <span class="math-container">$R$</span> to be symmetric, we would have to have <span class="math-container">$$y\ R\ x\qquad\text{whenever}\qquad x\ R\ y.$$</span> </p>
<p>According to the definition of <span class="math-container">$R$</span> tha... |
4,380,475 | <p>I'm trying to differentiate <span class="math-container">$x\sqrt{4-x^2}$</span> using the definition of derivative.</p>
<p>So it would be something like</p>
<p><span class="math-container">$$\underset{h\to 0}{\text{lim}}\frac{(h+x) \sqrt{4-\left(h^2+2 h x+x^2\right)}-x \sqrt{4-x^2}}{h}$$</span></p>
<p>I was trying t... | Andrew D. Hwang | 86,418 | <p>As Spivak points out in his <em>Calculus</em>, the proofs of limit theorems are strategies for implementing the definition of a limit.</p>
<p>Here, guided by the trick
<span class="math-container">\begin{align*}
\frac{f(x + h)g(x + h) - f(x)g(x)}{h}
&= \frac{f(x + h)g(x + h) - f(x + h)g(x) + f(x + h)g(x) - f... |
1,988,419 | <p>Any hint for proving this? If $Y$ is a subspace of $X$, what I am able to find is a closed subset $V$ in $Y$, hence $\mbox{cl}_Y(V)$ is compact, whose closure is contained in a neighborhood of a point $x$, by regularity of $Y$. Restricted to $X$, this $V_x=V \cap X$ is closed. I dont see any way to prove that $V_x$ ... | DanielWainfleet | 254,665 | <p>Let $X=[0,1]$ and $Y=X\cap \mathbb Q.$ Suppose $p\in U\subset Y$ where $U$ is a nbhd ,in $Y,$ of $p.$ Let $U\supset U'\cap Y$ where $U'$ is open in $X$ and $p\in U'.$ There exists $[a,b]\subset X$ with $a<b$ and $p\in [a,b]\cap Y$ and $[a,b]\subset U'.$ Now $$Cl_Y(U)=Y\cap Cl_X(U)\supset Y\cap Cl_X(U'\cap Y)\sup... |
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