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 |
|---|---|---|---|---|
2,483,231 | <p>If $F(x)=f(g(x))$, where $f(5) = 8$, $f'(5) = 2$, $f'(−2) = 5$, $g(−2) = 5$, and
$g'(−2) = 9$, find $F'(−2)$. I'm totally lost on this problem, I'm assuming to incorporate the Chain Rule. I get $5(5) * 9 = 225$ but I am incorrect.</p>
<p>Update: Thanks guys, I see where I messed up thanks!</p>
| ultrainstinct | 177,777 | <p>We have that $$F'(x) = f'(g(x))g'(x),$$
and since we know $g(-2)$, $g'(-2)$, and $f'(5)$, we can find the final answer of
$$F'(-2) = f'(g(-2))g'(-2) = f'(5)\cdot 9 = 18.$$</p>
|
1,643,649 | <blockquote>
<p>I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$.</p>
</blockquote>
<p>$\Bbb Z^*_n$ means integers up to $n$ coprime with $n$</p>
<p>I do not know how to do this. I have difficulties doing proofs involving isomorphisms. A methodological answer would be highly appreciated.</p>
<... | user26857 | 121,097 | <p>Order of $3$ in $\mathbb Z_{10}^*$ is $4$, while the order of elements from $\mathbb Z_8^*$ is $2$.</p>
|
121,362 | <p>I have a set of sample time-series data below of monthly prices for two companies. </p>
<p>Q1. I want to calculate monthly and quarterly log returns.what is the most expedient way to do this? <code>TimeSeriesAggregate[]</code> only has the standard <code>Mean</code>, etc. </p>
<p>Q2. With the returns from Q1, wha... | J. M.'s persistent exhaustion | 50 | <p>I guess something like this:</p>
<pre><code>With[{n = 7},
BlockRandom[SeedRandom["triangles"];
Graphics[Table[{RandomColor[],
RegularPolygon[{Sqrt[3] (j + i - 1),
3 j + Boole[EvenQ[i]]}/2,
... |
121,362 | <p>I have a set of sample time-series data below of monthly prices for two companies. </p>
<p>Q1. I want to calculate monthly and quarterly log returns.what is the most expedient way to do this? <code>TimeSeriesAggregate[]</code> only has the standard <code>Mean</code>, etc. </p>
<p>Q2. With the returns from Q1, wha... | Wjx | 6,084 | <p>This question is not a bit hard:</p>
<pre><code>mat = {{1, 0}, {1/2, Sqrt[3]/2}};
draw[n_] :=
Graphics[Table[{RandomColor[],
Triangle[{{i + n + 1 - #, j + n + 1 - #}, {i, j + 1}, {i + 1,
j}}.mat]}, {i, n}, {j, # - i}] & /@ {n, n + 1}];
draw[8]
</code></pre>
<p>Code is easy, check it by y... |
370,007 | <p>A river boat can travel a 20km per hour in still water. The boat travels 30km upstream against the current then turns around and travels the same distance back with the current. IF the total trip took 7.5 hours, what is the speed of the current? Solve this question algebraically as well as graphically..</p>
<p>I st... | Math Gems | 75,092 | <p><strong>Hint</strong> <span class="math-container">$\rm\ (ab)^k\! = 1\Rightarrow a^k\! = b^{-k}\! =\color{#c00}c \in \langle a\rangle\cap\langle b\rangle\Rightarrow ord\,c\mid m,n\,\Rightarrow\, ord\,c\mid(m,n)\!=\!1\,\Rightarrow\, \color{#c00}{c\! =\! 1},\,$</span> thus <span class="math-container">$\rm\ a^k\! = 1 ... |
1,356,367 | <p>Is it true that projection is a normal matrix? It's clear that orthogonal projection is, but what about non-orthogonal projection?</p>
<p>By normal matrix, I mean matrix A such that $AA' = A'A$.</p>
| quid | 85,306 | <p><a href="http://en.wikipedia.org/wiki/Binomial_distribution" rel="nofollow">From the relevant Wikipedia page:</a> </p>
<blockquote>
<p>The binomial distribution is frequently used to model the number of successes in a sample of size $n$ drawn with replacement from a population of size $N$. </p>
</blockquote>
<p>... |
1,018,248 | <p>Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like $\mathbb{P}[X$ has $n$ jumps in $[0,1]]$ or $\mathbb{P}[X$ has a jump of absolute value $>u$ in $[0,1]]$ for some $... | saz | 36,150 | <p>Yes, there is a very strong relation between the (distribution of the) jumps of a Lévy process and its Lévy measure. In fact, the Lévy measure describes the jump behaviour of the corresponding Lévy process:</p>
<p>Define the jump counting measure</p>
<p>$$N([0,t] \times B) := |\{0 \leq s \leq t; \Delta X_s \in B\}... |
2,825,789 | <p>I struggle to understand the following theorem (not the proof, I can't even validate it to be true). Note: I don't have a math background.</p>
<blockquote>
<p>If S is not the empty set, then (f : T → V) is injective if and only if Hom(S, f) is injective.</p>
<p>Hom(S, f) : Hom(S, T) → Hom(T, V)</p>
</blockquote>
<p>... | Paramanand Singh | 72,031 | <p>It is worthwhile to give another proof for Riemann integrability of functions which are continuous on a closed interval.</p>
<p>The proof below is taken from <em>Calculus</em> by Spivak and I must say it is novel enough. It does not make use of uniform continuity bur rather invokes mean value theorem for derivative... |
1,925,867 | <p>I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.</p>
| user259242 | 308,784 | <p>If $S$ is a subring of $R$ we write $S\hookrightarrow R$. The hooked arrow means 'monomorphism', which encapsulates the idea of $S$ being isomorphic to a sub-object of $R$ which is in this case a sub<em>ring</em>.</p>
<p><strong>EDIT</strong></p>
<p>After the discussion in quid's answer I will explain this a littl... |
3,792,135 | <p><strong>Question:</strong> Sum of the series <span class="math-container">$1-3x^2+5x^4 - ... + (-1)^{n-1} (2n-1)x^{2n-2} = \sum\limits_{n=0}^{\infty} (-1)^{n-1} (2n-1)x^{2n-2}$</span></p>
<p>My first idea is to integrate to get <span class="math-container">$\int f(x) dx = x -x^3 + x^5 - ... + (-1)^{ n-1}x^{2n-1} = \... | Riemann'sPointyNose | 794,524 | <p>You had the right idea. Now,</p>
<p><span class="math-container">$${\sum_{n=1}^{\infty}(-1)^{n-1}x^{2n-1}=\left(-\frac{1}{x}\right)\sum_{n=1}^{\infty} (-1)^{n}x^{2n}=\left(-\frac{1}{x}\right)\sum_{n=1}^{\infty}(-x^2)^{n}}$$</span></p>
<p>Now, we get that this is a Geometric Series with ratio <span class="math-contai... |
1,186,825 | <p>Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$</p>
<p>I tried, but failed. Any help will be appreciated.</p>
<p>At most points $(\cos 1/x)^n\to 0$, but how can I prove that the integral tends to zero clearly and convincingly?</p>
| Villetaneuse | 222,146 | <p>We know that $\left|\cos(\frac{1}{x})\right|<1$ except on a countable set, which hence has measure 0.</p>
<p>Therefore, for almost any $x \in [0;1]$, $\lim_{n \to +\infty} \left(\cos \frac{1}{x}\right)^n =0$.</p>
<p>Since for any $x \in [0,1]$ and any $n$, $|\cos(\frac{1}{x})^n|\leq 1$, we can conclude by using... |
1,186,825 | <p>Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$</p>
<p>I tried, but failed. Any help will be appreciated.</p>
<p>At most points $(\cos 1/x)^n\to 0$, but how can I prove that the integral tends to zero clearly and convincingly?</p>
| yakaqi | 220,822 | <p>you can try to substitute $t=\frac1x$.</p>
<p>$$\lim_{n\to\infty}\int_0^1\cos^n \frac1x\mathrm dx \Rightarrow \lim_{n\to\infty}\int_{+\infty}^1\ \frac {cos^n(t)}{-t^2}\mathrm dt \Rightarrow
\lim_{n\to\infty}\int_1^{+\infty}\ \frac {cos^n(t)}{t^2}\mathrm dt$$</p>
<p>Next we consider about the integral.</p>
<p>\beg... |
374,909 | <p>If <span class="math-container">$A\subseteq\mathbb{N}$</span> is a subset of the positive integers, we let <span class="math-container">$$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$</span> be the <em>upper density</em> of <span class="math-container">$A$</span>.</p>
<p>For <span class="math-con... | Random | 88,679 | <p>Notice that <span class="math-container">$\sigma(p^{k-1}) = k$</span> and so the image of <span class="math-container">$\sigma$</span> is all of <span class="math-container">$\mathbb{N}$</span>.</p>
<p>By the way, <span class="math-container">$\sigma$</span> is usually used for the sum of divisors function, and it i... |
3,029,208 | <p>Hi I have been trying to find a way to find a combinatorial proof for <span class="math-container">${kn \choose 2}= k{n \choose 2}+n^2{k \choose 2}$</span>. </p>
| Daniel Robert-Nicoud | 60,713 | <p><strong>Hint:</strong> You want to pick <span class="math-container">$2$</span> elements out of <span class="math-container">$k$</span> buckets of <span class="math-container">$n$</span> elements each. You have two possible ways to do it: either you pick a bucket, and then you take <span class="math-container">$2$</... |
2,895,284 | <blockquote>
<p>Find $\frac{d}{dx}\frac{x^3}{{(x-1)}^2}$</p>
</blockquote>
<p>I start by finding the derivative of the denominator, since I have to use the chain rule. </p>
<p>Thus, I make $u=x-1$ and $g=u^{-2}$. I find that $u'=1$ and $g'=-2u^{-3}$. I then multiply the two together and substitute $u$ in to get:</p... | Umberto P. | 67,536 | <p>No it isn't. When you apply the quotient rule you should be differentiating the denominator $v(x) = (x-1)^2$, not its reciprocal.</p>
|
4,243,344 | <blockquote>
<p><span class="math-container">${43}$</span> equally strong sportsmen take part in a ski race; 18 of
them belong to club <span class="math-container">${A}$</span>, 10 to club and 15 to club <span class="math-container">${C}$</span>. What is the
average place for (a) the best participant from club <span c... | Mike Earnest | 177,399 | <p>Instead of finding the probability of each place and doing <span class="math-container">$\sum k\,p(k)$</span>, you can use this trick.</p>
<p>When all the people are lined up in order of place, the ten people in the <span class="math-container">$B$</span> group will divide the line into <span class="math-container">... |
1,013,484 | <p>I've this function : $f(x,y)= \dfrac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8}$ for $(x,y)\ne (0,0)$ and $0$ for $(x,y)=(0,0)$</p>
<p>It's admits directional derivatives at the origin?</p>
| user72272 | 72,272 | <p>A function admits directional derivative at a point if its gradient $\nabla{f}$ exists at that point. The gradient of your function is given by,
$$\nabla{f}=\left(\begin{array}{cc} -\frac{2\, x\, y^4\, \left(2\, x^6 + 3\, x^4 - 2\, y^4 + 1\right)}{{\left(x^6 + x^2 + 2\, y^4\right)}^2} & \frac{4\, x^2\, y^3\, \le... |
1,013,484 | <p>I've this function : $f(x,y)= \dfrac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8}$ for $(x,y)\ne (0,0)$ and $0$ for $(x,y)=(0,0)$</p>
<p>It's admits directional derivatives at the origin?</p>
| MickG | 135,592 | <p>Let $\nu=(\cos\alpha,\sin\alpha)$ be any vector in the plane $\mathbb{R}^2$. Let us calculate the limit (which will depend on alpha) which is the d.d. with respect to $\nu$ of $f$:
$$\frac{\partial f}{\partial\nu}(0,0)=\lim_{t\to0^+}\frac{f(t\cos\alpha,t\sin\alpha)}{t}=\lim\frac{(1+t^2\cos^2\alpha)t^6\cos^2\alpha\si... |
4,192,869 | <p>What is the difference between a set being an element of a <span class="math-container">$\sigma$</span>-algebra compared to being a subset of a <span class="math-container">$\sigma$</span>-algebra?</p>
| user0102 | 322,814 | <p>Let <span class="math-container">$\Omega$</span> be a nonempty set. We say that a class of subsets of <span class="math-container">$\Omega$</span> denoted by <span class="math-container">$\Sigma$</span> is a <span class="math-container">$\sigma$</span>-algebra iff</p>
<ol>
<li><span class="math-container">$\Omega\in... |
205,479 | <p>There are $K$ items indexed $X_1, X_2, \ldots, X_K$ in the pool. Person A first randomly take $K_A$ out of these $K$ items and put them back to the pool. Person B then randomly take $K_B$ out of these $K$ items. What is the expectation of items that was picked by B but not taken by A before?</p>
<p>Assuming $K_A \g... | Community | -1 | <p>André's solution is the best one, of course. </p>
<p>But for the sheer fun of it, let's calculate the sum
\begin{equation}
E = \sum_{i=1}^{K_B} i \frac{{{K}\choose{K_A}}{{K_A}\choose{K_B - i}}{{K - K_A}\choose{i}}}{{{K}\choose{K_A}}{{K}\choose{K_B}}}
\end{equation}
First, cancel the common factor
$$E = \sum_{i} i ... |
6,431 | <p>I hate to sound like a broken record, but closing <a href="https://math.stackexchange.com/q/219906/12042">this question</a> as <em>not constructive</em> makes no sense to me. The canned explanation reads in relevant part:</p>
<blockquote>
<p>We expect answers to be supported by facts, references, or specific expe... | Noah Snyder | 8 | <p>One thing to keep in mind is that the software picks the reason that got a majority of the close votes. I agree that "not constructive" is not applicable to this question and is a little harsh. I voted as "too localized" as it didn't seem to me that this question is of interest to anyone not working on this exact ... |
1,617,269 | <p>Let X and Y be independent random variables with probability density functions
$$f_X(x) = e^{-x} , x>0$$
$$f_Y(y) = 2e^{-2y} , y>0$$</p>
<p>Derive the PDF of $Z_1 = X + Y$</p>
<p>other cases: $Z =min(X,Y)$ , $Z =1/Y^2 $ , $Z =e^{-2y} $ </p>
<p>Just considering the 1st part, I understand to go from the fac... | Em. | 290,196 | <p>It appears that
\begin{align*}
P(X+Y<z)&=\int_0^\infty\int_0^{-y+z} f_X(x)f_Y(y)\,dxdy\tag 1\\
&=\int_0^\infty\int_0^{-y+z} e^{-x}\cdot2e^{-2y}\,dxdy\\
&=\int_0^\infty 2e^{-2y}\left(1-e^{-(-y+z)}\right)\,dy\\
&=\int_0^\infty 2e^{-2y}\,dy-2e^{-z}\int_0^\infty e^{-y}\,dy\\
&=1-2e^{-z}
\end{align... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | mpiktas | 4,742 | <p>In <strong>Probability theory</strong> I was told by my professor, that that 3 most important theorems are Central Limit Theorem, Law of Large Numbers and Law of the Iterated Logarithm.</p>
|
3,009,387 | <p>I'm asking the following: it is true that if <span class="math-container">$K$</span> is a normal subgroup of <span class="math-container">$G$</span> and <span class="math-container">$K\leq H\leq G$</span> then <span class="math-container">$K$</span> is normal in <span class="math-container">$H$</span>? I tried to pr... | Peter Szilas | 408,605 | <p>Numerator</p>
<p><span class="math-container">$2x^2-50=2(x-5)(x+5)$</span>.</p>
<p>Denominator</p>
<p><span class="math-container">$2x^2+3x -35 =(2x-7)(x+ 5)$</span></p>
<p><span class="math-container">$\dfrac{2(x-5)(x+5)}{(2x-7)(x+5)}=$</span></p>
<p><span class="math-container">$\dfrac{2(x-5)}{2x+7}.$</span><... |
2,853,401 | <p>Assume $E\neq \emptyset $, $E \neq \mathbb{R}^n $. Then prove $E$ has at least one boundary point. (i.e $\partial E \neq \emptyset $).</p>
<p>================= </p>
<p>Here is what I tried.<br>
Consider $P_0=(x_1,x_2,\dots,x_n)\in E,P_1=(y_1,y_2,\dots,y_n)\notin E $.<br>
Denote $P_t=(ty_1+(1-t)x_1,ty_2+(1-t)x_2,\... | DanielWainfleet | 254,665 | <p>Take $p\in E.$ For $0\ne q\in \Bbb R^n$ let $S(q)=\{x\geq 0: \{p+yq: 0\leq y\leq x\}\subset E\}.$ </p>
<p>Take $q$ such that $z=\sup S(q)<\infty$. Such $q$ exists, otherwise $E=\Bbb R^n.$ </p>
<p>(i). If $p+zq\in E$ then $p+zq$ is in the closure of $E^c\cap \{p+yq:y>z\},$ which is a subset of $\overline {E... |
58,209 | <p>Question: Of the following, which is the best approximation of
$$\sqrt{1.5}(266)^{3/2}$$</p>
<p>$$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$</p>
<p>I used $1.5\approx1.44=1.2^2$ and $266\approx256=16^2$. Therefore the approximation by me is $4096$, so I chose $(D)$ which is wrong. The correct ... | anon | 11,763 | <p>$$\sqrt{1.5}\cdot266^{3/2}\approx1.2 \times 16^3 = 4915.2$$</p>
<p>The closest answer is (E) 5300. Great intuition on how to find simple approximations, but you forgot to multiply by $1.2$! Also note that $1.44<1.5$ and $256<266$, so you know the true answer must be above the discovered approximation, leaving... |
58,209 | <p>Question: Of the following, which is the best approximation of
$$\sqrt{1.5}(266)^{3/2}$$</p>
<p>$$(A)~1,000~~~~(B)~2,700~~~~(C)~3,200~~~~(D)~4,100~~~~(E)~5,300$$</p>
<p>I used $1.5\approx1.44=1.2^2$ and $266\approx256=16^2$. Therefore the approximation by me is $4096$, so I chose $(D)$ which is wrong. The correct ... | kuch nahi | 8,365 | <p>$$\sqrt{\frac{3}{2}} \cdot ( \sqrt{266})^3 =\sqrt{\frac{3\cdot 266}{2}}\cdot (\sqrt{266})^2 = \sqrt{399} \cdot 266 \approx 266 \cdot 20 = 5320$$</p>
<p>This is closest to option (E)</p>
<p><strong>Edit</strong>: Note that the only approximation I used here is $\sqrt{399}\approx \sqrt{400}$ so the result will dif... |
1,917,942 | <p>Prove that every homogeneous equation of second degree in $x$ and $y$ represents a pair of lines, each passing through the origin.</p>
<p>My Attempt:
Let $ax^2+2hxy+by^2=0$ be a homogeneous equation of second degree in $x$ and $y$.</p>
<p>We can write this equation as
$$by^2+2hxy+ax^2=0$$
Dividing both sides by $x... | coffeemath | 30,316 | <p>You have divided by $x^2,$ but no loss there since if $x^2=0$ then $x=0$ and that leads to $y=0$ provided $b \neq 0.$ In that case the point $(0,0)$ is on it.</p>
<p>If it happens that $b=0,$ then it factors as $x(a+2hy)=0,$ and setting each factor to zero gives a line through the origin as desired.</p>
<p>Now ass... |
1,917,942 | <p>Prove that every homogeneous equation of second degree in $x$ and $y$ represents a pair of lines, each passing through the origin.</p>
<p>My Attempt:
Let $ax^2+2hxy+by^2=0$ be a homogeneous equation of second degree in $x$ and $y$.</p>
<p>We can write this equation as
$$by^2+2hxy+ax^2=0$$
Dividing both sides by $x... | H. H. Rugh | 355,946 | <p>Writing $ax^2+2b xy +c y^2$ for the quadratic form, the number of lines depends upon the eigenvalues of the matrix</p>
<p>$$ \left(\begin{matrix} a & b \\ b & c \end{matrix} \right)$$</p>
<p>If the matrix is positive or negative definite there is only the origin when setting the form to zero (I assume we a... |
362,926 | <p>I have a problem that looks like this:</p>
<p>$$\frac{20x^5y^3}{5x^2y^{-4}}$$</p>
<p>Now they said that the "rule" is that when dividing exponents, you bring them on top as a negative like this:</p>
<p>$$4x^{5-2}*y^{3-(-4)}$$</p>
<p>That doesn't make too much sense though. A term like $y^{-4}$ is essentially say... | amWhy | 9,003 | <p>There is nothing wrong with your thinking. You are correct.</p>
<p>But so is the text: Note that you do indeed arrive at the same answer. </p>
<p>$$ 4x^{5-2}y^{3-(-4)}= 4x^{5-2}y^{3+4} = 4x^3y^7 $$</p>
<p>The textbook did exactly what you <em>both did</em> with $\dfrac{x^5}{x^2} = x^{5-2}$: subtracting the expon... |
4,345,671 | <p>I have a series of cubic polynomials that are being used to create a trajectory. Where some constraints can be applied to each polynomial, such that these 4 parameters are satisfied.
-Initial Position
-final Position
-Initial Velocity
-final Velocity</p>
<p>The polynomials are pieced together such that the ends of o... | bubba | 31,744 | <p>After suitable scaling, the sides of the pseudo-triangle shown in the answer from Andrew Hwang are the graphs of <span class="math-container">$\cos(x)$</span> and <span class="math-container">$-\cos(x)$</span> curves, over the range <span class="math-container">$0$</span> to <span class="math-container">$\tfrac12 \p... |
764,905 | <p>Calculate $$\int_{D}(x-2y)^2\sin(x+2y)\,dx\,dy$$ where $D$ is a triangle with vertices in $(0,0), (2\pi,0),(0,\pi)$.</p>
<p>I've tried using the substitution $g(u,v)=(2\pi u, \pi v)$ to make it a BIT simpler but honestly, it doesn't help much.</p>
<p>What are the patterns I need to look for in these problems so I ... | colormegone | 71,645 | <p>This problem seems to have been designed for the use of variable substitutions and a Jacobian determinant. <strong>Luka Horvat</strong>'s intuition is proper, and <strong>Santiago Canez</strong> makes the proposal, that substitutions <span class="math-container">$ \ u \ = \ x - 2y \ , \ v \ = \ x + 2y \ $</span... |
3,200,330 | <p>Suppose <span class="math-container">$\Phi: A\to A$</span> is a transformation of the set <span class="math-container">$A$</span>. I want to understand what it means for a subset <span class="math-container">$B\subseteq A$</span> to be invariant under <span class="math-container">$\Phi$</span>. </p>
<p><a href="htt... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$B \subset \Phi^{-1}(B)$</span> is equivalent to <span class="math-container">$\Phi (B) \subset B$</span>. Both say the same thing: whenever <span class="math-container">$b \in B$</span> we also have <span class="math-container">$\Phi (b) \in B$</span>.</p>
|
1,220,790 | <p>Consider the black scholes equation, </p>
<p>$$
\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0
$$</p>
<p>How do I show that if $V( S, t)$ is a solution, then $S(\frac{\partial V}{\partial S })$ is also a solution?</p>
<p>I... | Mark Joshi | 106,024 | <p>$$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0$$
We can regroup
$$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 \left( S\frac{\partial }{\partial S}\right)^2 V + ( r-q - \sigma^2/2 )S\frac{\partial V }{\partial S } ... |
1,220,790 | <p>Consider the black scholes equation, </p>
<p>$$
\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0
$$</p>
<p>How do I show that if $V( S, t)$ is a solution, then $S(\frac{\partial V}{\partial S })$ is also a solution?</p>
<p>I... | Danny | 203,396 | <p>Ok for the second part on $W(S,t)=S^\beta V(1/S,t)$, I've figured it out. Just some carelessness on my part when tossing terms around. It goes as follows...</p>
<p>The partial derivatives are,
$$\begin{align}
\frac{\partial W}{\partial S} & = \beta S^{\beta -1 }V- S^{\beta -2}\frac{\partial V}{\partial S}\\
\fr... |
58,772 | <p>Is there any general way to find out the coefficients of a polynomial.</p>
<p>Say for e.g.<br>
$(x-a)(x-b)$ the constant term is $ab$, coefficient of $x$ is $-(a+b)$ and coefficient of $x^2$ is $1$.</p>
<p>I have a polynomial $(x-a)(x-b)(x-c)$.
What if the number is extended to n terms.?</p>
| Gadi A | 1,818 | <p>Try opening it to get the feel for yourself; in general, the coefficient of $x^k$ is a sum of all the products on choices of $n-k$ out of the possible $n$ roots, multiplied by $(-1)^k$. So for $(x-a)(x-b)(x-c)(x-d)$ you'll get that the coefficient of $x^2$ is $ab+ac+ad+bc+bd+cd$.</p>
<p>For a precise discussion see... |
2,208,113 | <p>Let $x$ and $y \in \mathbb{R}^{n}$ be non-zero column vectors, from the matrix $A=xy^{T}$, where $y^{T}$ is the transpose of $y$. Then the rank of $A$ is ?</p>
<hr>
<p>I am getting $1$, but need confirmation .</p>
| quasi | 400,434 | <p>Suppose $x,y,z$ are real numbers such that
\begin{align*}
x + y &= \sqrt{4z-1}\\[4pt]
y + z &= \sqrt{4x-1}\\[4pt]
z + x &= \sqrt{4y-1}\\[4pt]
\end{align*}</p>
<p>Let $s = x + y + z$. Since $x + y \ge 0,\;\;y + z \ge 0,\;\;z + x \ge 0$, we have $s \ge 0$.
<p>
Then from the original system of equations, w... |
166,925 | <p>I have a function <code>u[y]</code> and I want to find the limit of integration that integration is equal zero.</p>
<pre><code>Λ = -30;
u[η_] := (2*η - 2*η^3 + η^4) + Λ/6*(η - 3*η^2 + 3*η^3 - η^4);
θ = Integrate[u[η]*(1 - u[η]), {, 0, 1}] // N;
δ = 1/θ;
u[y_] := Piecewise[{{1,y > δ}}, (2*y/δ - 2*(y/δ)^3 + (y/δ)^... | Community | -1 | <p><strong>Attempt 3.</strong></p>
<p>It turns out you can trick Mathematica into not complaining about lack of derivatives by adding a dependent derivative in a separate equation:</p>
<pre><code>sum3 = NDSolveValue[
{Laplacian[dummy[x, y], {x, y}] == 0,
s[x, y] == f1[x, y] + f2[x, y]
}, s... |
887,200 | <p>So I have the permutations:
$$\pi=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8
\end{array} \right)$$
$$\sigma=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9... | user1729 | 10,513 | <p>It depends.</p>
<p>Where I did my undergrad we learned to write permutations on the right, so $3\sigma\pi=\ldots$. This can be found in older books. The more common (people will claim, "standard") notation today is to write $\sigma\pi(3)$. Therefore, it depends on what your book or lecturer is telling you. If you h... |
2,536,553 | <p>I know that is possible to apply the spectral decomposition (diagonalization) to a matrix when the sum of the dimensions of its eigenspaces is equal to the size of the matrix.</p>
<p>The spectral decomposition is:</p>
<p>$$
F=P\Lambda P^{-1}
$$</p>
<p>where $\Lambda$ is the diagonal matrix of eigenvalues and $P$ ... | JayTuma | 506,755 | <p>You know that in a PID a element is irreducible if and only if it is prime. Then, if $p_1$ on the left divides $q_1 \cdot \ldots \cdot q_m$ then it must divide one of this factors, lets say $q_i$. But $q_i$ is irreducible, thus</p>
<p>$$ p_1 = uq_i $$</p>
<p>where $u$ is invertible. By induction, you can then show... |
1,413,145 | <p>I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing.</p>
<p>$\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for $n\ge6$.)</p>
<p>I was led to this problem by trying to prove by induction that $\big(1+\frac{1}{n}\big)^n\le3-\f... | Robert Israel | 8,508 | <p>Let
$$ \eqalign{f(n) = \dfrac{1}{n} + \left( 1 + \dfrac{1}{n}\right)^n &= \dfrac{1}{n} + \exp\left( n \ln\left(1+\dfrac{1}{n}\right)\right) \cr &=
\dfrac{1}{n} + \exp\left(1 - \dfrac{1}{2n} + \dfrac{1}{3n^2} + O\left(\dfrac{1}{n^3}\right)\right) \cr &= e - \dfrac{e-2}{2n} + \dfrac{11e}{24 n^2} + O\left(\... |
2,738,957 | <p>I did the following to derive the value of $\pi$, you might want to grab a pencil and a piece of paper:</p>
<p>Imagine a unit circle with center point $b$ and two points $a$ and $c$ on the circumference of the circle such that triangle $abc$ is an obtuse triangle. you can see that if $\theta$ denotes the angle $\an... | Mohammad Riazi-Kermani | 514,496 | <p>Your mathematics is good up to </p>
<p>$$\lim_{\theta \to 0} -\frac{\theta\pi}{90} = \lim_{\theta \to 0} \sin(2\theta)$$ which is simply $$0=0$$ which is the same as $$-0=0$$ </p>
<p>You want to manipulate $$0=0$$ by dividing both side by $0$ to generate the undefined $$0/0$$ and get a negative value for $\pi $</p... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | Jacob Akkerboom | 4,330 | <p><strong>Level one version</strong></p>
<p>This is an adaptation of amr's answer (based on Kuba's answer)</p>
<pre><code>mapAtLevOneIndexed[f_, list_, pos_] :=
ReplacePart[list,
Inner[Rule[#, f[#2, #]] &, pos, Part[list, pos], List]]
</code></pre>
<p>Example</p>
<pre><code>mapAtLevOneIndexed[f, {1, 2... |
2,337,583 | <p>I cannot understand the inductive dimension properly. I read something on Google but mostly there only are conditions or properties. Not a definition. I got to know about it from the book “ The fractal geometry of nature”. ( I am a 12 grader.)</p>
| Lehs | 171,248 | <p>The small inductive dimension can be defined inductively by </p>
<ol>
<li><p>$\text{ind}(\emptyset)=-1$ </p></li>
<li><p>$\text{ind}(\{x\})=0$ </p></li>
<li><p>$\text{ind}(X)$ is the smallest number $n$ such that for all $x\in X$ and every open set $U\ni x$ there is an open set $V\ni x$ with $\bar{V}\subseteq U$... |
81,715 | <p>I am a graduate student in physics trying to learn differential geometry on my own, out of a book written by Fecko.</p>
<p>He defines the gradient of a function as:</p>
<p>$
\nabla f = \sharp_g df = g^{-1}(df, \cdot )
$</p>
<p>This makes enough sense to me. However, when I try to calculate the gradient of a fu... | Marián Fecko | 19,396 | <p>In vector calculus one usually computes in terms of <strong>orthonormal</strong> (rather than <strong>coordinate</strong>) components of vector fields.
This stuff is discussed in detail a bit later in the book :-)
See pages 183-4 (Section 8.5.)
You find explicit expressions in both components, there.
Good luck!</p>... |
1,192,338 | <p>How to prove that there are infinite taxicab numbers?
ok i was reading this <a href="http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers" rel="nofollow">http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers</a>
and thought of this question..any ideas?</p>
| Joshua Tilley | 389,601 | <p>Use Ramanujan's identity that
<span class="math-container">$$\left(x^2+7xy-9y^2\right)^3+\left(2x^2-4xy+12y^2\right)^3=\left(2x^2+10y^2\right)^3+\left(x^2-9xy-y^2\right)^3$$</span></p>
<p>Reference:</p>
<p><a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html" rel="nofollow noreferrer">http://ma... |
78,423 | <p>How far can one go in proving facts about projective space using just its universal property?</p>
<p>Can one prove Serre's theorem on generation by global sections, calculate cohomology, classify all invertible line bundles on projective space?</p>
<p>I don't like many proofs of some basic technical facts very aes... | Daniel Litt | 6,950 | <p>I agree with Anton that it would be too much to hope for to get serious results (e.g. cohomology of line bundles) from the "nice" universal property of projective space, but one can indeed prove that there are no non-constant regular functions on $\mathbb{P}^n$ using only the universal property. </p>
<p>Namely, it... |
899,230 | <p>It seems that both isometric and unitary operators on a Hilbert space have the following property:</p>
<p><span class="math-container">$U^*U = I$</span> (<span class="math-container">$U$</span> is an operator and <span class="math-container">$I$</span> is an identity operator, <span class="math-container">$^*$</spa... | Jonas Meyer | 1,424 | <p>An isometric operator on a (complex) Hilbert space is a linear operator that preserves distances. That is, $T$ is an isometry if (by definition) $\|Tx-Ty\|=\|x-y\|$ for all $x$ and $y$ in the space. By linearity, this is equivalent to $\|Tx\|=\|x\|$ for all $x$. Because of the definition of the norm in terms of ... |
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Ovi | 64,460 | <p>With L'Hopital's rule:</p>
<p>$\lim_{x\to1}\large \frac{\frac {d}{dx} (x^{1/5}-1)}{\frac {d}{dx} (x^{1/3}-1)}=\lim_{x\to1} \large\frac { \frac 15 x^{-4/5}}{\frac 13 x^{-2/3}}$</p>
<p>Since we are dividing, we subtract the exponents of $x$ and get:</p>
<p>$$ \lim_{x \to 1} \frac 35 x^{(-\frac 45)- (-\frac 23)}=\li... |
435,079 | <p>This is exercise from my lecturer, for IMC preparation. I haven't found any idea.</p>
<p>Find the value of</p>
<p>$$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$</p>
<p>Thank you</p>
| Start wearing purple | 73,025 | <p>Mathematica evaluates the integral to
$$\int_0^{1}(1+x^n)^{1/n}dx={}_2F_1\left(-\frac{1}{n},\frac1n,1+\frac1n;-1\right).\tag{1}$$
Next, let us write the standard series representation for the hypergeometric function
$$_2F_1(a,b,c;t)=\sum_{k=0}^{\infty}\alpha_kt^k,\qquad \alpha_k=\frac{\Gamma(a+k)\Gamma(b+k)\Gamma(c)... |
1,681,205 | <p>I would like a <strong>hint</strong> for the following, more specifically, what strategy or approach should I take to prove the following?</p>
<p><em>Problem</em>: Let $P \geq 2$ be an integer. Define the recurrence
$$p_n = p_{n-1} + \left\lfloor \frac{p_{n-4}}{2} \right\rfloor$$
with initial conditions:
$$p_0 = P ... | mjqxxxx | 5,546 | <p>Consider the vector $X_n = (p_n, p_{n-1}, p_{n-2}, p_{n-3})$. We have
$$
\begin{eqnarray}
X_{n+1} &=& (p_{n+1},p_n, p_{n-1},p_{n-2}) \\
&=& \left(p_n+\left\lfloor \frac{1}{2}p_{n-3}\right\rfloor, p_n, p_{n-1}, p_{n-2}\right) \\ &=& (p_n+ \frac{1}{2}p_{n-3}, p_n, p_{n-1}, p_{n-2}) - (\varepsi... |
1,746,180 | <p>I already solved a few integrals with substitution but in this case I have no idea how to start. How to solve the integral $$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$where $x=\frac{\pi}{2}-t$ with substitution, can you tell me how to start?
It would be great!</p>
| GoodDeeds | 307,825 | <p>$$\int_a^b f(x) dx=\int_a^b f(a+b-x) dx$$
$$\tag1I=\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$$
Replace $x$ by $\frac{\pi}2-x$.
$$I=\int_0^{\pi/2} \frac{\sqrt{\sin(\frac{\pi}2-x)}}{\sqrt{\sin(\frac{\pi}2-x)}+\sqrt{\cos(\frac{\pi}2-x)}} dx,$$
$$\tag2I=\int_0^{\pi/2} \frac{\sqrt{\cos(x)}}{... |
1,275,461 | <p>I am trying to proof that $-1$ is a square in $\mathbb{F}_p$ for $p = 1 \mod{4}$. Of course, this is really easy if one uses the Legendre Symbol and Euler's criterion. However, I do not want to use those. In fact, I want to prove this using as little assumption as possible.</p>
<p>What I tried so far is not really... | Mathmo123 | 154,802 | <p>For $p\ge 3$, $\mathbb F_p^*$ is a cyclic group of order $p-1$. If $g$ is a generator, then $g^\frac{p-1}2= -1$. In particular $-1$ will be a square if and only if $\frac{p-1}{2}$ is even - i.e. if $p \equiv 1 \pmod 4$.</p>
<p>Note that this proof isn't fundamentally different from those using Fermat's little theor... |
158,662 | <p>I know to prove a language is regular, drawing NFA/DFA that satisfies it is a decent way. But what to do in cases like</p>
<p>$$
L=\{ww \mid w \text{ belongs to } \{a,b\}*\}
$$</p>
<p>where we need to find it it is regular or not. Pumping lemma can be used for irregularity but how to justify in a case where it can... | Boris Trayvas | 33,272 | <p>An alternative way of proving a language is regular/irregular is the <a href="http://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem" rel="nofollow">Myhill-Nerode theorem</a>.</p>
|
1,246,705 | <p>I was doing some linear algebra exercises and came across the following tough problem :</p>
<blockquote>
<p>Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\phi:M_{n\times n}(\mathbf{R})\to M_{n\times n}(\mathbf{R})$ is a nonzero linear transform (i.e. t... | user1551 | 1,551 | <p>Here is an alternative proof that is much less computational. It works over any field. Let $E_{ij}$ be the matrix with a $1$ at the $(i,j)$-th entry and zeros elsewhere.</p>
<ol>
<li><p>$\phi$ is injective and hence an automorphism. Suppose the contrary that $\phi(A)=0$ for some nonzero $A$. Since $A$ is nonzero, e... |
4,598,275 | <p>Let <span class="math-container">$(X, \mathcal{F})$</span> be a measurable space, and <span class="math-container">$\mu_{n}, \mu$</span> probability measures on it. <span class="math-container">$\mu_{n}$</span> is said to converge weakly to <span class="math-container">$\mu$</span> if for any bounded continuous func... | Tom | 986,425 | <p>After discussing the problem with others I'm now trying to show the progress I make by far.</p>
<p>Suppose <span class="math-container">$g \in C_{b}(\mathbb{R}^{d})$</span>. We want to show</p>
<p>(1): <span class="math-container">$\int g d\mu_{n} \xrightarrow{} \int g d\mu$</span>.</p>
<p>Now fix any <span class="m... |
489,109 | <p>I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials.</p>
<p>Please note: This is an intro to calculus, so we haven't learned derivatives or anything too complex.</p>
<p>Here's the question: </p>
<p>Let $f(x) = x^5 + x + 7$. Find the value of the inverse function a... | davidlowryduda | 9,754 | <p><strong>HINT(s)</strong></p>
<ol>
<li>$f$ is an increasing function.</li>
<li>Since $f$ is increasing, you will be able to modify your guesses to close in on the answer quickly.</li>
</ol>
|
1,501,876 | <blockquote>
<p>I want to prove $A_n$ has no subgroups of index 2. </p>
</blockquote>
<p>I know that if there exists such a subgroup $H$ then $\vert H \vert = \frac{n!}{4}$ and that $\vert \frac{A_n}{H} \vert = 2$ but am stuck there. I have tried using the proof that $A_4$ has no subgroup of order 6 to get some idea... | David Hill | 145,687 | <p>As has been stated in the comments, if you know that $A_n$ is simple for $n\geq 5$, and subgroups of index 2 are normal, you are done with that case. </p>
<p>For the $n=4$ case, you do need to do a bit more work. Your idea to show that $A_n$ has no subgroup of order 6 is correct. Well, a subgroup of order $6$ in $A... |
3,168,119 | <p>How do I solve for n?</p>
<p><span class="math-container">$125 = x * 2^n$</span></p>
<p>This is what I have so far:</p>
<p><span class="math-container">$5^3 = x * 2^n$</span></p>
<p>I do remember that according to the exponential rules,
that the powers should be the same if the equation is like this:</p>
<p><sp... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: We get <span class="math-container">$$\frac{125}{x}=2^n$$</span> so <span class="math-container">$$\ln\left(\frac{125}{x}\right)=n\ln(2)$$</span></p>
|
4,139,141 | <p>If the conditions of the theorem are met for some ordinary differential equation, then we are guaranteed that a solution exists. However, I don't fully understand what it means for a solution to exist. If we can show that a solution exists, does that mean that it can be found explicitly using known methods? Or, are ... | Henno Brandsma | 4,280 | <ol>
<li><p>The <span class="math-container">$V_a$</span> are open for each <span class="math-container">$a \in A$</span> (that's how it's chosen in the application of Hausdorffness) and each <span class="math-container">$a \in A$</span> is covered by "its own" <span class="math-container">$V_a$</span>, so it... |
14,238 | <p>In question #7656, Peter Arndt asked <a href="https://mathoverflow.net/questions/7656/why-does-the-gamma-function-complete-the-riemann-zeta-function">why the Gamma function completes the Riemann zeta function</a> in the sense that it makes the functional equation easy to write down. Several of the answers were from... | Kevin Buzzard | 1,384 | <p>I am no expert, but let me give you some guesses as to the answers.</p>
<p>Q1) I am going to go for "no". I think it's precisely Tate's thesis that shows that the gamma factor for Riemann zeta can be interpreted as a local factor analogous to the usual local factors at the finite primes. However let me absolutely s... |
115,367 | <p>Let $f(z)$ be an analytic function on $D=\{z : |z|\leq 1\}$. $f(z) < 1$ if $|z|=1$. How to show that there exists $z_0 \in D$ such that $f(z_0)=z_0$. I try to define $f(z)/z$ and use Schwarz Lemma but is not successful. </p>
<p>Edit: Hypothesis is changed to $f(z) < 1$ if $|z|=1$. I try the following. If $f$ ... | davin | 2,881 | <p>Fixed point theorem should suffice, but here's a slightly more complex-analysis-style proof:</p>
<p>By Rouche's theorem, we note that $|f(z)|<1=|z|$ on $\{|z|=1\}$ therefore $N_{f(z)-z}(D)=N_z(D)=1$, so we've proven something slightly stronger; that there exists <em>a unique</em> $z_0 \in D$ such that $f(z_0)-z_... |
795,193 | <p><strong>THEOREM</strong>: Suppose $\{f_n\}$ is a sequence of continuous functions from $[a,b]$ to $\Bbb R$ that converge pointwise to a continuous function $f$ over $[a,b]$. If $f_{n+1}\leq f_n$, then convergence is uniform. </p>
<p>Then, why is the continuity of the the functions $f_i$'s important for the theorem?... | Sameer Kailasa | 117,021 | <p>As far as I know, there isn't a way to prove this theorem without using properties of compactness. In fact, compactness of domain is necessary for the statement to be true.</p>
<p>So, let's just develop those relevant properties here, in $\mathbb{R}$ instead of in metric spaces. Hopefully working through these defi... |
215,983 | <p>I was expecting to get the answer to the above by doing either of the following:</p>
<pre><code>Options[Graph[{1 -> 2, 2 -> 1}], GraphLayout]
AbsoluteOptions[Graph[{1 -> 2, 2 -> 1}], GraphLayout]
</code></pre>
<p>However the result to both is</p>
<blockquote>
<p>GraphLayout->Automatic</p>
</blockquo... | Szabolcs | 12 | <p>I believe that <code>GraphLayout -> Automatic</code> typically resolves to one of the following:</p>
<ul>
<li>For large graphs, the default is <code>"SpringElectricalEmbedding"</code>.</li>
<li><p>Small undirected trees up to 49 vertices use <code>"LayeredEmbedding"</code>. It looks like this:</p>
<p><a href="h... |
539,363 | <p>Two horses start simultaneously towards each other and meet after $3h 20 min$. How much time will it take the slower horse to cover the whole distance if the first arrived at the place of departure of the second $5 hours$ later than the second arrived at the departure of the first.</p>
<p><strong>MY TRY</strong>::<... | lab bhattacharjee | 33,337 | <p>Let the distance from the meeting place of the departure of the first Horse $A$ is $a $ meter
and the distance from the meeting place of the departure of the first Horse $B$ is $b$ meter</p>
<p>So, the total distance is $a+b$ meter</p>
<p>So, the speed of the first horse is $\displaystyle\frac a{200}$ meter/min... |
3,118,462 | <p>cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent. </p>
<p>What is the probability that <strong>at least</strong> 3 cars arrive before a truck arrives? </p>
<p>My thoughts:
Interarrival of cars A ~ Exp(2 p... | fleablood | 280,126 | <p>You have to guess. But use some tricks.</p>
<p><span class="math-container">$68 = 2^2*17$</span> so the only options are <span class="math-container">$1$</span> and <span class="math-container">$2^2*17=68$</span>, or <span class="math-container">$2$</span> and <span class="math-container">$2*17 = 34$</span>, or <s... |
9,462 | <p>In my question I ask for practical tips for the mathematical research practice, if not personal,I look for some articles/websites/books/guides/faq related, or if was already asked on Math.SE, the link to the question.</p>
<p><a href="https://math.stackexchange.com/questions/386520/practical-tips-research-and-discov... | Tom Oldfield | 45,760 | <p>I imagine people voted to close since the question would be seen by many to be off topic, I think that it is certainly borderline.</p>
<p>However, I think that it is valid and appropriate for the site. Most questions are related to specific mathematical problems, in the sense of proving things or answering question... |
4,417,896 | <p>I have only found information regarding doing this by integration by parts. By differentiating under the integral sign, I let
<span class="math-container">$$I_n = \int_0^\infty x^n e^{-\lambda x} dx $$</span>
and get <span class="math-container">$\frac{dI_n}{d\lambda} = -I_{n+1} $</span> and therefore <span class="m... | Dr. Sundar | 1,040,807 | <p>Let us define
<span class="math-container">$$
I_n = \int\limits_{x = 0}^\infty \ x^n \, e^{-\lambda x} \ dx \tag{1}
$$</span></p>
<p><strong>Method 1: Using Gamma Functions</strong></p>
<p>Use the substitution
<span class="math-container">$$
\lambda x = t \ \ \mbox{or} \ \ x = {t \over \lambda} \tag{2}
$$</span></p>... |
3,345,329 | <p>In Bourbaki Lie Groups and Lie Algebras chapter 4-6 the term displacement is used a lot. For example groups generated by displacements. But I can not find a definition of the term displacement given anywhere. I also looked at Humphreys Reflection Groups and Coxeter groups book but I could not find it. Can someone pr... | River Li | 584,414 | <p>WLOG, assume that <span class="math-container">$\sigma = 1$</span>.</p>
<p>We have a random sample <span class="math-container">$X_1, X_2, \cdots, X_n$</span> of size <span class="math-container">$n$</span> from <span class="math-container">$X\sim N(0, 1)$</span>.
The probability density function of <span class="ma... |
3,921,847 | <p>I had the following question:</p>
<blockquote>
<p>Does there exist a nonzero polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfying both of the following conditions?</p>
<ul>
<li><span class="math-container">$P(x)$</span> has no rational root;</li>
<li>For every positive integer <... | JMP | 633,430 | <p>Consider the polynomial</p>
<p><span class="math-container">$$
P(x) = (x^2 - 13)(x^2 - 17)(x^2 - 221)
$$</span></p>
<p>Clearly this has no rational solutions. We wish to show that the congruence <span class="math-container">$P(x) \equiv 0 \bmod{m}$</span> is solvable for all integers <span class="math-container">$m$... |
891,370 | <p>I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.</p>
| MPW | 113,214 | <p><strong>Hint:</strong> Use the fact that $\log(ab^c)=\log a + c\log b$</p>
|
16,795 | <p>Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:</p>
<ol>
<li>as a logical formula $\Phi= \bigwedge_{i,j\in[n]} \neg_{ij}\ Rx_ix_j$ with $\neg_{ij} = \neg$ or $ \neg\neg$ </li>
<li>as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$ </li>
</ol>
... | Joel David Hamkins | 1,946 | <p>Perhaps what Hans means is simply that any graph has exactly the same information as the complement graph, because if we know completely where there are no edges, then we also know completely where are the edges, and conversely.</p>
<p>But having the same information in this logical sense is not the same as being i... |
4,580,717 | <p>I know I can express "everyone is A" as:</p>
<p>P: is a person
<span class="math-container">$$ \forall x (Px \implies Ax) $$</span></p>
<p>And I can express "everyone who's A is B" as:</p>
<p><span class="math-container">$$ \forall x ((Px \land Ax) \implies Bx) $$</span></p>
<p>But how can I expr... | Bram28 | 256,001 | <p>"Everyone who's A" is not a sentence, because it has no truth-value.</p>
|
3,383,687 | <p>I'm interested in ideas for improving and fixing the proof I wrote for the following theorem:</p>
<blockquote>
<p>Let <span class="math-container">$f \colon \mathbb{R}^n \to \mathbb{R} $</span> be differentiable, and <span class="math-container">$ \lim_{\| x \| \to \infty} f(x) = 0 $</span>. Then <span class="mat... | user284331 | 284,331 | <p>Suppose <span class="math-container">$f$</span> is not identically zero, say, <span class="math-container">$f(x_{0})\ne 0$</span>, choose a big <span class="math-container">$M>0$</span> such that <span class="math-container">$|x_{0}|<M$</span> and <span class="math-container">$|f(x)|<|f(x_{0})|$</span> for ... |
3,434,242 | <p>If I need to get some variables values from a vector in Matlab, I could do, for instance, </p>
<pre><code>x = A(1); y = A(2); z = A(3);
</code></pre>
<p>or I think I remember I could do something like</p>
<blockquote>
<p>[x, y, z] = A;</p>
</blockquote>
<p>However Matlab is not recognizing this format. what w... | horchler | 80,812 | <p>@Thales is correct. If <code>A</code> happens to be a <a href="https://www.mathworks.com/help/matlab/matlab_prog/what-is-a-cell-array.html" rel="nofollow noreferrer">cell array</a> rather than a matrix you could do something like this:</p>
<pre><code>A = {1 2 3};
[x,y,z] = A{:}
</code></pre>
<p>But that is as clos... |
4,510,697 | <p><a href="https://i.stack.imgur.com/HKvBy.png" rel="nofollow noreferrer">How do I find explicit solutions of x and y in this system?</a></p>
| on1921379 | 805,886 | <p>There's a simpler solution if you consider <span class="math-container">$\frac{x^6 - 1}{x - 1} = (x^2 + x + 1)(x^3 + 1)$</span>. Let <span class="math-container">$d = \gcd(x^2 + x + 1, x^3 + 1)$</span>, then <span class="math-container">$d \mid x^2 + x + 1 \mid x^3 - 1$</span> and <span class="math-container">$d \mi... |
958,099 | <p>I have the following question:</p>
<blockquote>
<p>For real $x$, $f(x) = \frac{x^2-k}{x-2}$ can take any real value. Find the range of values $k$ can take.</p>
</blockquote>
<p>Here is how I commenced:</p>
<p>$$
y(x-2) = x^2-k \\
-x^2 + xy - 2y + k = 0\\
$$</p>
<p>So we have $a=-1$, $b=y$, $c=(-2y+k)$. In orde... | najayaz | 169,139 | <p>Your last inequality was: $$y^2-8y+4k\ge0$$
For it to be true for all real values of y, the discriminant of the LHS should always be non-positive. So we have:$$64-16k\le0$$$$16k\ge64$$$$k\ge4$$</p>
|
646,183 | <p>I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation.</p>
<p>One can find the real and complex parabolic, elliptic, hyperbolic, subgroups, $\mathrm{SU}(2)$, $\mathrm{SU}(1,1)$ and $\mathrm... | Moishe Kohan | 84,907 | <p>I will start with the general strategy and the will show how to use it in the case of $SL(2,C)$. Let $G$ be a connected Lie subgroup of another Lie group $H$; in your case, $H$ is a complex Lie group, which helps. You first look at the Levi-Malcev decomposition of the Lie algebra $G$ and ask if the solvable radical ... |
3,095,710 | <p>Factor <span class="math-container">$x^8-x$</span> in <span class="math-container">$\Bbb Z[x]$</span> and in <span class="math-container">$\Bbb Z_2[x]$</span></p>
<p>Here what I get is <span class="math-container">$x^8-x=x(x^7-1)=x(x-1)(1+x+x^2+\cdots+x^6)$</span> now what next? Help in both the cases in <span clas... | Lubin | 17,760 | <p>To answer a question of yours in the comments, here’s how I thought in factoring <span class="math-container">$(x^7-1)/(x-1)$</span> over <span class="math-container">$\Bbb F_2$</span>:</p>
<p>You’re still talking about the six primitive seventh roots of unity here. But what is the smallest field containing <span c... |
275,785 | <p>Let $a_{1}, a_{2}, \ldots, a_{n}$, $n \geq 3$. Prove that at least one of the number $(a_{1}+a_{2}\ldots +a_{n})^{2}-(n^2-n+2)a_{i}a_{j}$ is greater or equal with $0$ for $1 \leq i < j \leq n$.</p>
<p>I don't know at least how to catch this problem .
Thanks :)</p>
| Bojan Serafimov | 56,860 | <p>suppose it's not true, add everything together.</p>
<p>$t = \frac{n(n-1)}{2}$</p>
<p>$A = \sum a_i^2$</p>
<p>$B = \sum a_ia_{i+1}$</p>
<p>After adding you will get</p>
<p>$tA < 2B$</p>
<p>Which is a contradiction because $t > 2$ and $A \geq B$.</p>
|
231,583 | <p>Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry. </p>
<p>Rather than only logic and elementary geometry, are there existing research results which by using machine learning techniques(possibly generative learning models) to disco... | Noah Stein | 5,963 | <p>One example that is close to, if not exactly of this type, is <a href="http://arxiv.org/abs/1601.07227">Veit Elser's demonstration</a> that machine learning techniques can learn how to do fast matrix multiplication from examples of matrix products. As far as I know this method has so far just reproduced some known ... |
231,583 | <p>Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry. </p>
<p>Rather than only logic and elementary geometry, are there existing research results which by using machine learning techniques(possibly generative learning models) to disco... | Tadashi | 22,389 | <p>A pre-print by Li-An Yang, Jui-Bin Liu, Chao-Hong Chen and Ying-ping Chen was submitted to arXiv last month (24 Feb) giving some preliminary results of an ad-hoc evolutionary algorithm used to prove some simple theorems within the Coq proof assistant:
<a href="http://arxiv.org/abs/1602.07455" rel="nofollow noreferre... |
4,501,286 | <blockquote>
<p>If a, b, c are positive real numbers such that <span class="math-container">$a^2+ b^2+ c^2 = 1$</span>
<br>Show that:
<span class="math-container">$$\frac{1}{a} +\frac{1}{b} +\frac{1}{c}+a +b +c \geq 4\sqrt{3}.$$</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>First , I used Holder's : <... | Dr. Mathva | 588,272 | <p>Alternative approach using the <em>point of incidence technique</em>.</p>
<p>We will first employ the AM-GM inequality, followed by QM-AM:
<span class="math-container">\begin{align*}
a+b+c+\frac1a+\frac1b+\frac1c&=a+b+c+\frac13\left(\frac1a+\frac1b+\frac1c\right) + \frac23\left(\frac1a+\frac1b+\frac1c\right)\\
&... |
2,357,899 | <p>I want to prove that the cartesian product of a finite amount of countable sets is countable. I can use that the union of countable sets is countable. </p>
<p><strong>My attempt:</strong></p>
<p>Let $A_1,A_2, \dots, A_n$ be countable sets. </p>
<p>We prove the statement by induction on $n$</p>
<p>For $n = 1$, th... | Community | -1 | <p>For countable $A_1,...,A_m$ form their cartesian product $A_1 \times ... \times A_m=\{(a_1,...,a_m) ; a_1 \in A_1,...,a_m \in A_m\}$.</p>
<p>This cartesian product can be written as $\bigcup_{j_1=1}^{w_1}... \bigcup_{j_m=1}^{w_m} \{(a_{j_1 1}...a_{j_m m} )\}$ where $w_k$ is either finite or equal to $\infty$ for $k... |
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | Community | -1 | <p>Your equation is: $\{4 n\}^\frac32 = \frac12$.(I think you understood this)</p>
<p>Now, write $4=2^2$ in the left side. Then the equation looks like</p>
<p>$(2^2)^\frac32 \times n^\frac32=\frac12$ </p>
<p>$\Rightarrow $$n^\frac32 =$$\frac1{16}$</p>
<p>$\Rightarrow n=(\frac1{16})^\frac23$$=\frac14\times 2^{-\frac... |
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | k170 | 161,538 | <p>$$4n^{\frac32} = 8^{-\frac{1}{3}}= \frac{1}{8^{\frac{1}{3}}}$$
$$4\sqrt{n^3}= \frac{1}{\sqrt[3]{8}}= \frac{1}{2}$$
$$\sqrt{n^3}= \frac{1}{8}$$
$$\left(\sqrt{n^3}\right)^2= \left(\frac{1}{8}\right)^2$$
$$\left|n^3\right|= \frac{1}{8^2}=\frac{1}{64}$$
Note that $n^{\frac32}$ or $\sqrt{n^3}$ implies that $n^3\geq 0$ an... |
2,843,560 | <p>If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to</p>
<p>(a) $y$</p>
<p>(b) $y/2$</p>
<p>(c) $2y$</p>
<p>(d) $y/6$</p>
<p>I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: $$\sin(x)+\sin82x)+\sin(3x)=\sin(2x)(2\cos(x)+1)=\sin(y)$$</p>
<p>$$\cos(x)+\cos(2x)+\cos(3x)=\cos(2x)(2\cos(x)+1)=\cos(y)$$
from here you will get</p>
<p>$$\tan(2x)=\tan(y)$$
con you finish?</p>
|
529,708 | <p>A is prime greater than 5, B is A*(A-1)+1,if B is prime,</p>
<p>then digital root of A and B must the same.(OEIS A065508)</p>
<p>Sample: 13*(13-1)+1 = 157
13 and 157 are prime and have same digital root 4</p>
| Empy2 | 81,790 | <p>There are two cases: $A=3k-1$ and $A=3k+1$.<br>
Expand $A*(A-1)+1$ in both these cases.<br>
$B=9k^2-9k+3$ or $B=9k^2+3k+1$</p>
|
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Confutus | 40 | <p>I have come to understand this to be analogous to an order relationship among the truth values. P -> Q should be understood to mean "Q is at least as true as P", or "Q is not less true than P". </p>
<p>So, any statement at all (Q = t or f) is at least as true as a known falsehood (P=f), and a known truth (Q=t) is a... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Brendan W. Sullivan | 80 | <p>I find it helpful to introduce the negation of conditional claims simultaneously. For one, this better helps them to understand the "false implies false" case; but also, this helps them understand how to logically negate conditional claims (which is essential when they go on to learn proof techniques for conditional... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Pete L. Clark | 176 | <p>Various psychological studies have been done which show that most people (including university students, who are the most common subjects of psychological tests!) are very poor at grappling with the last two entries of the truth table for <span class="math-container">$A \implies B$</span> in an abstract context, but... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | mbork | 704 | <p>I am really impressed by the answers given by others here; I will definitely keep them in mind when teaching freshmen next semester. But I also have my 2 cents to add, since I haven't seen anything like that in them. I realize that this is only some kind of vague intuition, and it would probably confuse a lot of s... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | zhw. | 228,045 | <p>Every derivative you ever saw is an example. Suppose $f'(c)$ exists. Let $g(x) = (f(x) - f(c))/(x-c).$ Then $\lim_{x\to c} g(x) = f'(c).$ But $c$ is not in the domain of $g.$ This is the primordial example.</p>
|
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | tparker | 268,333 | <p>While this might be controversial, I'll make the claim that no physical quantity $f(x)$ can ever usefully be thought of as having a removable singularity. By definition, a physical quantity must be physically measurable, and every measurement has an associated error. The probability (not probability density, but abs... |
3,133,328 | <p>I am currently working on a question about proving the sum of eigenvalues and I have been searching for the solution <a href="https://www.youtube.com/watch?v=OLl_reBXY-g" rel="nofollow noreferrer">from YouTube</a>.</p>
<p>However, I do not understand why the teacher uses the diagonal method to show that the sum of ... | R. W. Prado | 284,531 | <p>There is an known easy proof for that. Indeed, let A be an <span class="math-container">$n \times n$</span> matrix, due <a href="https://www.google.com/search?q=Schur%20decomposition&oq=Schur%20decomposition&aqs=chrome..69i57j69i59l2j69i60&sourceid=chrome&ie=UTF-8" rel="nofollow noreferrer">Schur dec... |
570,740 | <p>Hi there I'm having some trouble with the following problem:</p>
<p>I have a $3\times3$ symmetric matrix
$$
A=\pmatrix{1+t&1&1\\ 1&1+t&1\\ 1&1&1+t}.
$$
I am trying to determine the values of $t$ for which the vector $b = (1,t,t^2)^\top$ (this is a column vector) is in the column space of $A$... | user1337 | 62,839 | <p>In case you are familiar with determinants, you can see that the matrix is invertible, unless $t \in \{0,-3 \}$. If $A$ is invertible its column space is all of $\mathbb R^3$, and the two remaining cases $t=0,-3$ are easy to check separately.</p>
|
570,740 | <p>Hi there I'm having some trouble with the following problem:</p>
<p>I have a $3\times3$ symmetric matrix
$$
A=\pmatrix{1+t&1&1\\ 1&1+t&1\\ 1&1&1+t}.
$$
I am trying to determine the values of $t$ for which the vector $b = (1,t,t^2)^\top$ (this is a column vector) is in the column space of $A$... | LinAlgMan | 49,785 | <p>Here is the matrix
$$ A = \left[ \begin{matrix} 1+t & 1 & 1 \\ 1 & 1+t & 1 \\ 1 & 1 & 1+t \end{matrix} \right] $$
and denote
$$ b = \left[ \begin{matrix} 1 \\ t \\ t^2 \end{matrix} \right] $$
Then $b$ is in the column space of $A$ if and only if there is
$$ v = \left[ \begin{matrix} x \\ y ... |
570,740 | <p>Hi there I'm having some trouble with the following problem:</p>
<p>I have a $3\times3$ symmetric matrix
$$
A=\pmatrix{1+t&1&1\\ 1&1+t&1\\ 1&1&1+t}.
$$
I am trying to determine the values of $t$ for which the vector $b = (1,t,t^2)^\top$ (this is a column vector) is in the column space of $A$... | egreg | 62,967 | <p>Gaussian elimination is not difficult in this case:
\begin{align}
\left[\begin{array}{ccc|c}
1+t & 1 & 1 & 1 \\
1 & 1+t & 1 & t \\
1 & 1 & 1+t & t^2
\end{array}\right]
&\to
\left[\begin{array}{ccc|c}
1 & 1 & 1+t & t^2 \\
1 & 1+t & 1 & t \\
1+t & 1 &... |
623,703 | <blockquote>
<p>Find the exact value of $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )$ without using a calculator. </p>
</blockquote>
<p>I started by finding $\sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )=\dfrac{\pi}{4}$</p>
<p>So, $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\r... | Michael Albanese | 39,599 | <p><strong>Hint:</strong> $$\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)}$$</p>
|
10,666 | <p>My question is about <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">nonstandard analysis</a>, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of <em>the</em> nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question... | John Goodrick | 93 | <p>I think that the nonstandard models of R* will be fairly wild by most reasonable metrics, since the theory is <a href="http://en.wikipedia.org/wiki/Stable_theory" rel="nofollow">unstable</a> (the universe is linearly ordered). For instance, I don't think that arbitrary models will be determined up to isomorphism by... |
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