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 |
|---|---|---|---|---|
2,151,937 | <p>Let $A$ be an $m\times n$ real matrix, $x$ an $n\times 1$ vector and $b$ an $m\times 1$ vector. I want to compute
\begin{equation}
\dfrac{\partial }{\partial x} \Vert Ax+b\Vert^{2}.
\end{equation}
First, I expanded
\begin{equation}
\Vert Ax+b\Vert^{2}=(Ax+b)^{T}(Ax+b)=x^{T}A^{T}Ax+2x^{T}A^{T}b+b^{T}b
\end{equation}
... | Logan Stapleton | 399,630 | <p>A general way of finding the gradient <span class="math-container">$\nabla_x f(x)$</span> of any vector-valued function <span class="math-container">$f:\mathbb{R}^n \rightarrow \mathbb{R}$</span> is by using Taylor's Theorem, which can be expressed as such:
<span class="math-container">$$f(x+\delta) = f(x) + [\nabla... |
2,939,163 | <p>I want to find a certain <span class="math-container">$x$</span> that belongs to <span class="math-container">$\mathbb R$</span> so that </p>
<p><span class="math-container">$$\left|\begin{array}{r}1&x&1\\x&1&0\\0&1&x\end{array}\right|=1$$</span></p>
<p>This should be easy enough. I apply t... | Peter Szilas | 408,605 | <p>Polynomial long division:</p>
<p><span class="math-container">$\small{(x^3-2x+1)÷(x-1)= x^2 +x-1}$</span>;</p>
<p><span class="math-container">$\small{ -(x^3-x^2)}$</span></p>
<p><span class="math-container">$-----$</span></p>
<p><span class="math-container">$\small{x^2-2x +1}$</span></p>
<p><span class="math-c... |
3,715,475 | <p>About a year ago I asked <a href="https://math.stackexchange.com/questions/3258617/alaoglu-theorem-over-the-p-adics">here</a> whether the Banach-Alaoglu Theorem works over the <span class="math-container">$p$</span>-adics. The satisfactory answer I got is that the "usual" proof only uses local compactness, and so th... | Robert Furber | 184,596 | <p>Chilote has pointed to the right notion for the general case. I will answer the literal question.</p>
<p>The Banach-Alaoglu theorem (using the usual topological notion of compactness) cannot hold for normed spaces over a field with valuation <span class="math-container">$(k,|\cdot|)$</span> if the unit ball of <span... |
1,554,603 | <p>Let $\theta \in \mathbb R$, and let $T\in\mathcal L(\mathbb C^2)$ have canoncial matrix</p>
<p>$M(T)$ = $$
\left(
\begin{matrix}
1 & e^{i\theta} \\
e^{-i\theta} & -1 \\
\end{matrix}
\right)
$$
(a) Find the eigenvalues of $T$... | lhf | 589 | <p>The ideals of $\mathbb Z/60$ correspond to the ideals of $\mathbb Z$ that contain $60\mathbb Z$ and so correspond to the divisors of $60$. Since $60=2^2\cdot 3 \cdot 5$, it has $(2+1)\cdot(1+1)\cdot(1+1)=12$ divisors.</p>
|
3,989,921 | <p>Answered (I think!):</p>
<p>The triple product's purpose is to find a direction to the origin, perpendicular to the baseline, which is super trivial in 2D as there is only two perpendicular orientations, but the "cylinder" distinction is made in 3D because there are infinite perpendicular orientations - he... | Z Ahmed | 671,540 | <p>Vector trple product is defined as <span class="math-container">$$\vec P \times (\vec Q \times \vec R)= (\vec P \cdot \vec R)\vec Q-(\vec P\cdot \vec Q) \vec R.$$</span> Then
<span class="math-container">$$\vec V=(\vec A \times \vec B) \times \vec A=-\vec A \times (\vec A \times \vec B)=-(\vec A \cdot \vec B)\vec A... |
46,905 | <p>I need to draw a set of curves on one graph (characteristics equations). As you can see they have exchanged x and y axes. My goal is to plot all those curves on one graph. Are there ways to do that? </p>
<pre><code>f[t_, t0_] := -(2 - 4/Pi*ArcTan[2])*Exp[-t]*(t - t0);
g[x_, x0_] := (x - x0)/(-(2 - 4/Pi*ArcTan[x + ... | kglr | 125 | <p>Kuba's answer using <code>ParametricPlot</code> is the most convenient way to get the result you need. Alternatively, you can use a geometric transformation function that rotates and then reflects <code>s2</code> around the vertical axis:</p>
<pre><code>f[t_, t0_] := -(2 - 4/Pi*ArcTan[2])*Exp[-t]*(t - t0);
g[x_, x0... |
3,113,083 | <p>Why does every CNF formula for <span class="math-container">$(x_{1} \vee y_{1}) \wedge (x_{2} \vee y_{2})\wedge \ldots \wedge (x_{n} \vee y_{n})$</span> have at least <span class="math-container">$2^{n}$</span> terms?</p>
<p>This statement is on the Wikipedia page for DNF form here: <a href="https://en.wikipedia.or... | dan_fulea | 550,003 | <p>The combinatorial problem is (relatively) too complicated, and writing some code is (really) too simple, so here is the code and the count:</p>
<pre><code>S = SymmetricGroup(6)
R = Zmod(6)
cansPermutations = \
[ s for s in S
if not {R(0), R(1), R(-1)}.intersection(
{R(s(k)-1) - R(k-1) for k... |
2,994,296 | <p>I'm trying to figure out how to prove, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{4n}}{(4n)!} = 0$$</span>
The problem is, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{n}}{n!} = \infty$$</span>
and I have no idea how to prove the first limit equals <span class="math-container"... | Martín-Blas Pérez Pinilla | 98,199 | <p>Starting like other two answers:
<span class="math-container">$$a_n = \frac{n^{4n}}{(4n)!}\implies
\frac{a_{n+1}}{a_n} = \left(\left(1+\frac1n\right)^n\right)^4\cdot\frac{(n+1)^4}{(4n+1)(4n+2)(4n+3)(4n+4)}\to\frac{e^4}{4^4} < 1,
$$</span>
and by the quotient test <span class="math-container">$\sum a_n$</span> con... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Mark Bennet | 2,906 | <p>The probabilities work because there is a chance that more than one person is successful at the same time, even though there is also a chance that none are successful. The average number of successes for six people is six times the average for one person, but this average covers the case where all succeed at the sam... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Martin Argerami | 22,857 | <p>I think the situation is easier to see in a simpler example. Drop a coin, "heads" is success. The probability of success is 50%. This does <strong>not</strong> mean that if you drop the coin three times your probability of getting one head is 150%.</p>
<p>The probability of getting at least one head is one minus th... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | JKreft | 457,797 | <p>The problem is your third paragraph, where you've confused the expected number of successes in 6 tries (1.2) with a percentage chance (120%). This is one reason whey probability students are encouraged to work in decimals/fractions instead of percentages.</p>
<p>If you let six people try it, you can expect 1.2 succ... |
2,860,360 | <p>It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book <em>Algebraic Number Theory</em>:</p>
<p>"<em>Example 7.20</em> For $K=\mathbb{Q}(\sqrt[3]{2} )$, the discriminant is 108, and $r_{2}=1$. So the Minkowski bound is $\approx 2... | Acccumulation | 476,070 | <p>You've currently phrased this is terms of whether the probability is multiplicative (does having six times the number of trials give six times the probability of success), but we can equivalently ask whether it's additive (is the probability of success over two trials equal to the sum of probabilities for each indiv... |
2,880,830 | <p><a href="https://i.stack.imgur.com/a4Wh2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/a4Wh2.png" alt="enter image description here"></a></p>
<p>Not sure how to do this one. If <span class="math-container">$S$</span> is a field, then I was considering that <span class="math-container">$\exists ... | Alex Wertheim | 73,817 | <p>Uncover the spoilers for solutions completing the hints below:</p>
<ul>
<li><p>Suppose <span class="math-container">$R$</span> is a field, and let <span class="math-container">$s \in S$</span> be a nonzero element. Then multiplication by <span class="math-container">$s$</span> is an <span class="math-container">$R$... |
3,844,448 | <p>Find all values of <span class="math-container">$h$</span> such that rank(<span class="math-container">$A$</span>) = <span class="math-container">$2$</span>.</p>
<p><span class="math-container">$A$</span> = <span class="math-container">$\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\en... | Matthew H. | 801,306 | <p>Notice the plane <span class="math-container">$9x+y+3z=0$</span> is precisely the span of the first and third column of your matrix. All you need to do is find <span class="math-container">$h$</span> so that <span class="math-container">$(h,-1,1)$</span> resides on this plane; this will guarantee that you matrix has... |
79,084 | <p>Let $X$ be a topological space (say a manifold). A result of R. Thom states that the pushforwards of fundamental classes of closed, smooth manifolds generate the rational homology of $X$. This work of Thom predates the development of bordism. Is there now a more elementary proof of this result that does not rely ... | Sergey Melikhov | 10,819 | <p>A nice, direct combinatorial construction was given by Gaifullin, see his <a href="https://arxiv.org/abs/0712.1709" rel="nofollow noreferrer">papers</a> <a href="https://arxiv.org/abs/0806.3580" rel="nofollow noreferrer">on the</a> <a href="https://arxiv.org/abs/0912.3933" rel="nofollow noreferrer">arXiv</a> (equiva... |
1,967,847 | <blockquote>
<p>A vector space $V$ is called <strong>finite-dimensional</strong> if there is a finite subset of $V$ that is a basis for $V$. If there is no such finite subset of $V$, then $V$ is called <strong>infinite-dimensional</strong>.</p>
<hr>
<p>We now establish some results about finite-dimensional ... | copper.hat | 27,978 | <p>For each $n$, the following is a basis for $\mathbb{R}^2$, ${\cal B}_n = \{ (0,1), (n,1) \}$.</p>
<p>So, each ${\cal B}_n$ has exactly two elements, but there is an infinite number of bases.</p>
|
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... | amWhy | 9,003 | <p>Yes, you are correct in your thoughts in the last paragraph. </p>
<p>We use the chain rule:</p>
<p>Let $u = -2x.\;$ Then $\;\dfrac {du}{dx} = -2.$</p>
<p>This gives us $$\frac{d}{dx}(e^{-2x}) = \frac d{dx}(e^u) = e^u\left(\dfrac{du}{dx}\right) = e^{-2x}(-2) = -2e^{-2x}$$</p>
|
888,319 | <p><span class="math-container">$ABC$</span> is an acute angled triangle, where <span class="math-container">$P$</span> is the orthocenter, and <span class="math-container">$R$</span> is the circumradius. I want to show that <span class="math-container">$PA+PB+PC\le 3R$</span> geometrically, that is without using trig... | Jack D'Aurizio | 44,121 | <p>Yes, there is a way. Due to the Euler theorem, $O,G,H$ are collinear and $HO=3\,GO$. </p>
<p>This implies that if we take $O$ as the origin, the vector equation:
$$ H = A+B+C $$
holds. By applying the triangular inequality:
$$ OH = |H| \leq |A|+|B|+|C| = OA+OB+OC = 3R.$$</p>
|
1,572,593 | <p>Let $G$ a group, $H$ and $K$ two subgroups of G of finite order such that $H \cap K = \{1_G\}$. </p>
<p>I already show the first exercise which says that the cardinal of $HK$ is $|H||K|$.</p>
<p>The second exercise ask to deduce that if $|G|=pm$ where $p$ is a prime number and $p>m$, then $G$ has at most one su... | Justpassingby | 293,332 | <p>No need for sophistication. If $G$ has two distinct subgroups of order $p$ then those subgroups satisfy the conditions on $H$ and $K.$</p>
<p>Now since the subgroup is the only one of its order, it is identical to all of its conjugates and therefore normal.</p>
|
1,335,640 | <p>1) A disease has hit a city. The percentage of the population infected $t$ days after the disease arrives is approximated by $$p(t) = 12te^{\frac{-t}{7}} \qquad \mbox{for} \qquad0\leq t \leq 35.$$ </p>
<p>After how many days is the percentage of infected people a maximum? What is the maximum percent of the popu... | BruceET | 221,800 | <p><strong>Mode of gamma distribution.</strong> For (1), recognize that $p(x)$ is the PDF of a gamma distribution with shape parameter 2 and scale parameter 7. Look at the Wikipedia article. Its mode is $x = (2-1)7 = 7.$ (The bound 35 is so far out as to be irrelevant to the discussion.)</p>
<p><strong>Application of ... |
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... | André Nicolas | 6,312 | <p>We have
$$a_n=-\left(\frac{n^3+5}{n^2-8n}\right).$$
Informally, we now want to show that that if $n$ is "large" then $\dfrac{n^3+5}{n^2-8n}$ is large. We do not want to worry about negative values of $n^2-8n$, so suppose that $n\gt 8$, making $n^2-8n\gt 0$. For $n \gt 8$, we have
$$\frac{n^3+5}{n^2-8n} \gt \frac{n^... |
2,915,685 | <blockquote>
<p>Let $X$ be Banach space and $Y$ be a normed vector space and suppose that I find a linear map $T \in L(X,Y)$. With the aid of $T$, I wonder under what condition I can conclude that $Y$ is also a Banach space.</p>
</blockquote>
<p>Is $T$ a linear isomorphism enough? I encounter this question when thin... | detnvvp | 85,818 | <p>If $X$ is a Banach space, $Y$ is a linear space and $T:X\to Y$ is a linear isomorphism, then $Y$ is a Banach space. To show this let $(y_n)$ be a Cauchy sequence in $Y$, then the sequence $(T^{-1}y_n)$ is Cauchy in $X$, so it converges to some $x$, and then $y_n\to Tx$. Therefore $Y$ is complete.</p>
<p>Now, with t... |
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... | Theo Johnson-Freyd | 78 | <p>If you are interested in manifolds, then you might be interested in various related notions of <em>measures</em> and <em>distributions</em>. Let $M$ be a smooth manifold with algebra of functions $C^\infty(M)$. There is a very general notion of a <em>distribution</em>, which is that of any linear function $C^\inft... |
1,749,909 | <p>I need a help with somthing:
I need to tell if these two integrals are Convergence\Absolute convergence:</p>
<p>$\int _1^{\infty }\frac{\ln x}{\sqrt{x^3-x^2-x+1}}dx$, $\int _0^{\infty \:}\:\frac{\left(x^{\frac{1}{4}}+1\right)\cdot \sin\left(2\sqrt{x}\right)}{x}dx$
Now I compute this and I find that both converge. B... | jst345 | 327,162 | <p>Regarding your proof, to be honest, I think the student is right to be sceptical as there are a couple of issues with it! Firstly your suggestion that it is 'clear visually' that $\int^1_0 f(x) dx -\int^1_0 g(x) dx$ represents the area between the curves may very well be the case, but unfortunately when it comes to... |
2,278,431 | <p>"Apply Green's Theorem to evaluate the line integral of F around positively oriented boundary"</p>
<p>$$F(x,y)=x^2yi+xyj$$</p>
<p>C: The region bounded by y=$x^2$ and y=4x+5</p>
| Community | -1 | <p>Use the fact that $(1+\frac 1 n)^n$ <a href="https://math.stackexchange.com/questions/167843/i-have-to-show-1-frac1nn-is-monotonically-increasing-sequence"> is increasing</a> and $\lim (1+\frac1n)^n = e$ therefore $(1+\frac1n)^n \lt e$ and $\frac{u_{n+1}}{u_n} \lt 1$</p>
|
2,278,431 | <p>"Apply Green's Theorem to evaluate the line integral of F around positively oriented boundary"</p>
<p>$$F(x,y)=x^2yi+xyj$$</p>
<p>C: The region bounded by y=$x^2$ and y=4x+5</p>
| Clement C. | 75,808 | <p>The sequence is <strong>non-increasing</strong>. Before proving it, first: <em>compute the first terms, it will help you check whether your intuition is correct!</em></p>
<p>On <code>Mathematica</code> or <a href="http://www.wolframalpha.com/input/?i=DiscretePlot%5B+n%5En%2F(n!+E%5En),+%7Bn,+1,+100%7D%5D&t=crmt... |
207,807 | <p>Is there an explicit example of a non-commutative monoid $M$ such that for all elements $m,n \in M$ and $p \in \mathbb{N}$ we have $(m \cdot n)^p=m^p \cdot n^p$?</p>
<p>It suffices to construct a semigroup $H$ with an absorbing element $0$ such that $a^2=0$ for all $a$, because then $M := H \cup \{e\}$ will do the ... | Berci | 41,488 | <p>Let $M:=\{1,a,b\}$ with the following multiplication:
$$\begin{align} a\cdot b:=a & \quad b\cdot a:=b \\
a\cdot a:= a & \quad b\cdot b:= b \end{align} $$
(In other words, $xy=x$ holds for all $x\ne 1$, so associativity comes easily.)</p>
|
2,713,201 | <p>How would you work something like this out? </p>
<p>Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?</p>
| Community | -1 | <p>The go-to way for the derivative of $f^g$ is $(f(x)^{g(x)})'=(e^{g(x)\ln f(x)})'$ and then use chain rule. Or knowing by heart the formula that follows directly from this procedure. Of course, when $f(x)>0$, because for some reason we appear to be in a moment where people seem to be very adamant about saying stuf... |
2,713,201 | <p>How would you work something like this out? </p>
<p>Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?</p>
| The Integrator | 538,397 | <p>For any function of the type $u(x) ^{v(x)} $, use the logarithmic rule</p>
<p>ie, let $y =u^v$</p>
<p>$\implies ln(y) = vln(u) $</p>
<p>Differentiate wrt x</p>
<p>$\frac{dy}{dx} \frac 1y = v'ln(u) +\frac vu u'$</p>
<p>$\frac{dy}{dx} =y\big(v'ln(u) +\frac vu u'\big) $</p>
<p>For $ cos(x)^{cos(x)} $</p>
<p>$\fr... |
2,713,201 | <p>How would you work something like this out? </p>
<p>Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?</p>
| Latin Wolf | 536,550 | <p>So $\cos(x)^{\cos(x)}=e^{\cos(x)ln(\cos(x))}$ and $$\frac{d}{dx} e^{\cos(x)ln(\cos(x))}=e^{\cos(x)ln(\cos(x)} \cdot (-\sin(x) ln(\cos(x))+ -\sin(x)=\cos(x)^{\cos(x)} \cdot -\sin(x) ln(\cos(x))-\sin(x)$$</p>
|
3,443,094 | <blockquote>
<p>If <span class="math-container">$$\lim_{x\to 0}\frac{ae^x-b}{x}=2$$</span> the find <span class="math-container">$a,b$</span></p>
</blockquote>
<p><span class="math-container">$$
\lim_{x\to 0}\frac{ae^x-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)+a-b}{x}=\lim_{x\to 0}\frac{a(e^x-1)}{x}+\lim_{x\to 0}\frac{a-b}{... | trancelocation | 467,003 | <p>If you plug <span class="math-container">$a=b$</span> into your starting expression you get</p>
<p><span class="math-container">$$\lim_{x\to 0}a\frac{e^x-1}{x} = a \left.\left(e^x \right)'\right|_{x=0}= a \stackrel{!}{=} 2$$</span></p>
<p>Btw, you can show <span class="math-container">$a=b$</span> a bit easier by ... |
3,408,846 | <p>This is an example in Serge Lang "Introduction to Linear Algebra", page 48. I try to multiply these two <span class="math-container">$2$</span>x<span class="math-container">$3$</span> and <span class="math-container">$3$</span>x<span class="math-container">$2$</span> matrices but fail to obtain the result as mention... | NoChance | 15,180 | <p>My answer is a different. Notice the entry <span class="math-container">$a_{22}$</span> in the result.
<a href="https://i.stack.imgur.com/vq2aS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vq2aS.png" alt="enter image description here"></a></p>
|
3,811,154 | <p>Prove that this integral is less than infinity. If <span class="math-container">$0<a<c$</span> and <span class="math-container">$0<b$</span>: <span class="math-container">$$\int_0^\infty \frac{|x|^a}{(x+b)^{c+1}} dx.$$</span></p>
<p>From inspection, because <span class="math-container">$a<c$</span> and <... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Near <span class="math-container">$ +\infty $</span>, the integrand function satisfies</p>
<p><span class="math-container">$$\frac{x^a}{(x+b)^{c+1}}\sim \frac{1}{x^{c+1-a}}$$</span></p>
<p>So, if</p>
<p><span class="math-container">$$c+1-a>1$$</span>
or
<span class="math-container">$$... |
619,890 | <p>I have a question.There is a group of 5 men and a group of 7 women.With how many ways can each of the 5 men get married with one of the 7 women?</p>
| miniBill | 33,238 | <p>The projection of a point $x$ onto $L(S)$ is the intersection of $x + W^\bot$ with $L(S)$, where $W$ is the linear part of $L(S)$ and $W^\bot$ is it's orthogonal space, that is, the linear space of vector orthogonal to $W$.</p>
|
984,558 | <p>Is it possible to find a representation of the infinitesimal generators of the special unitary group SU(3) that contains 4 by 4 matrices, by say taking a Kronecker product of its irreducible representation(s) with itself?</p>
<p>I know this is possible for SU(2), where one can express the three 4 by 4 matrices span... | Stephen | 146,439 | <p>You might have a look at Fulton's book <em>Young tableaux</em> for the combinatorics relevant for computing these dimensions. Here is the relevant fact: the irreducible representations of $SU(n)$ may be indexed by partitions with at most $n-1$ parts in such a way that the dimension of the irreducible $L(\lambda)$ 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... | linksideal | 171,582 | <p>Note that you don`t want to prove or contradict $P$ or $R$ themselves. What you want to prove is the implication $P\rightarrow\neg R$. So you pick the assumption of this implication, in this case $P$, and use it for further argumentation.<br>
Now you can start your argumentation with another assumption like $R$ or w... |
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... | MattClarke | 118,792 | <p>Let me give a brief and less formal explanation than the other answers.</p>
<p>If you start with a set of assumptions {A<sub>1</sub>, A<sub>2</sub>, ... A<sub>n</sub>} and derive a contradiction, it proves that the set of assumptions is internally inconsistent (or as @J-Marcos phrased it, "jointly incompatible") --... |
357,520 | <p>How can I show that
$$\lim_{a\to{0}}\frac{{\pi}a\ \coth{\mathrm{{\pi}a}-1}}{2a^2}=\frac{\pi^2}{6}$$
I think the limit is in $\frac{0}{0}$ form, so I am using L'Hospital's rule, and then I cannot solve further, Please Help.</p>
<p>Thanks!</p>
| Community | -1 | <p>Let $t = \pi a$. We then get that
$$\dfrac{\pi a \coth(\pi a)-1}{2a^2} = \pi^2 \dfrac{t \coth(t)-1}{2t^2}$$
Now let us look at $f(t) = \dfrac{t \coth(t)-1}{2t^2}$. We then have
$$f(t) = \dfrac{\coth(t)}{2t} - \dfrac1{2t^2}$$
Now use the Laurent series expansion. $$\coth(t) = \dfrac1t + \dfrac{t}3 + \mathcal{O}(t^3)$... |
183,749 | <p>Considering two functions <span class="math-container">$\psi_{1}(u,v)$</span> and <span class="math-container">$\psi_{4}(u,v)$</span>. we have these two parial differential equation for them</p>
<p><span class="math-container">$(-2 i Sech[\frac{u}{\alpha}] \frac{\partial\psi_{4}}{\partial u}+2 i Sech[\frac{v}{\alp... | bbgodfrey | 1,063 | <p>This system of equations also can be solved symbolically, although not with <code>DSolve</code>. Beginning with </p>
<pre><code>eq = {((-2 I Sech[u/α] D[ψ4[u, v], u] + 2 I Sech[v/α] D[ψ4[u, v], v]) +
(-m ψ1[u, v] + 2 I Sech[u/α] D[ψ1[u, v], u] + 2 I Sech[v/α] D[ψ1[u, v], v])),
(( 2 I Sech[u/α] D[ψ1[u... |
4,039,424 | <p>I would like to calculate the elements of <span class="math-container">$\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$</span>.
I know that the elements of <span class="math-container">$\mathbb{Q}(\sqrt[3]{2})$</span> have the form of <span class="math-container">${a+b\sqrt[3]{2}+c\sqrt[3]{4}}$</span>, where a,b,c <span class="ma... | reuns | 276,986 | <p><span class="math-container">$$\sum_{p|n} 1/p \le \sum_{p| \prod_{q\le k_n} q}1/p= \sum_{p\le O(\log n)} 1/p= O(\log\log\log n)$$</span></p>
<p>where <span class="math-container">$\prod_{q\le k_n} q$</span> is the least primorial <span class="math-container">$\ge n$</span>:</p>
<p><span class="math-container">$\prod... |
3,478,098 | <p><a href="https://i.stack.imgur.com/dcxhi.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dcxhi.jpg" alt="enter image description here" /></a></p>
<p>Guys, Why is this weird statement true? It seems counterintuitive to me I cannot understand or lack creativity understanding it can you help me expla... | Eric Wofsey | 86,856 | <p>This is immediate from the definition of <span class="math-container">$\delta$</span>. If <span class="math-container">$x$</span> is a rational number with denominator at most <span class="math-container">$N$</span>, that means <span class="math-container">$x\in Q_N$</span>, so by definition <span class="math-conta... |
33,993 | <p>I am given the parameters for a bivariate normal distribution ($\mu_x, \mu_y, \sigma_x, \sigma_y,$ and $\rho$). How would I go about finding the Var($Y|X=x$)? I was able to find E[$Y|X=x$] by writing $X$ and $Y$ in terms of two standard normal variables and finding the expectation in such a manner. I am unsure how t... | Did | 6,179 | <p>Rather than embarking on some pretty involved computations of conditional distributions, one should rely on one of the main assets of Gaussian families, namely, the... </p>
<blockquote>
<p><strong>Key feature:</strong> In Gaussian families, conditioning acts as a linear projection.</p>
</blockquote>
<p>Hence, as... |
3,185,226 | <p>I have a simple question that confuses me for a while:</p>
<blockquote>
<p><span class="math-container">$$f(X) = \text{tr} \left( [ \log(X) ]^2 \right)$$</span></p>
<p>where <span class="math-container">$X$</span> is an <span class="math-container">$m \times m$</span> symmetric positive definite (SPD) matrix and <sp... | Robert Israel | 8,508 | <p>Careful: most of the standard calculus formulas for differentiation require things to commute. Matrices don't. So <span class="math-container">$d(Z^2)$</span> is not <span class="math-container">$2 Z \; dZ$</span>, it's <span class="math-container">$Z \; dZ + (dZ)\; Z$</span>. And I don't think there is a closed-... |
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>
... | Jack D'Aurizio | 44,121 | <p>We have $B_1 = \frac{1}{1+\lambda} A +\frac{\lambda}{1+\lambda} C$ and cyclic identities.<br>
We have $B_2 = \frac{\lambda}{1+\lambda} A_1 +\frac{1}{1+\lambda} C_1$ and cyclic identities.<br>
By combining them
$$ B_2 = \frac{\lambda}{1+\lambda}\left(\frac{1}{1+\lambda}C+\frac{\lambda}{1+\lambda}B\right)+\frac{1}{1+\... |
325,588 | <p>Is there an analytic function $f$ in $\mathbb{C}\backslash \{0\}$ s.t. for every $z\ne0$: $$|f(z)|\ge\frac{1}{\sqrt{|z|}}\, ?$$</p>
| MBM | 64,739 | <p>How about this:</p>
<p>Since $f(z)$ is analytic on $\mathbb{C}-\{0\}$, $g(z) = \frac{1}{(f(z))^2}$ is analytic on
$\mathbb{C}-\{0\}$. Also $\bigg|\frac{g(z)}{z}\bigg| \leq 1$. </p>
<p>I am sure you will finish the rest (think about the order of the pole at $0$ and use Liouville's Theorem).</p>
|
252,820 | <p><code>Sound[]</code> generates a visual representation of notes. I would like to extract that image, only notes, without controls and borders. How can I do it?</p>
<p>Take this example</p>
<pre><code>Sound[SoundNote @@@ Transpose @ {
{"E5", "D5", "F#4", "G#4", "C#5"... | creidhne | 41,569 | <p>Use <a href="http://reference.wolfram.com/language/ref/SparseArray.html" rel="nofollow noreferrer"><code>SparseArray</code></a> and <a href="http://reference.wolfram.com/language/ref/Band.html" rel="nofollow noreferrer"><code>Band</code></a> to reproduce your example 6-by-6 matrix:</p>
<pre><code>m = SparseArray[
... |
2,804,495 | <p>I was asked to solve this double integral:
Compute the area between $y=2x^2$ and $y=x^2$ and the hyperbolae $xy=1$ and $xy=2$ in </p>
<p>$$ \iint dx \,dy$$</p>
<p>I tried to solve it starting with considering that </p>
<p>$$x^2 \leq y \leq 2x^2 $$</p>
<p>suitabile for integration interval in $y$, obtaining the i... | mechanodroid | 144,766 | <p>$y \in [x^2, 2x^2]$ can be interpreted as $\frac{y}{x^2} \in [1,2].$</p>
<p>Consider the change of variables</p>
<ul>
<li><p>$u = \frac{y}{x^2} \in [1,2]$</p></li>
<li><p>$v = xy \in [1,2]$</p></li>
</ul>
<p>The Jacobian is given by</p>
<p>$$\frac1J = \begin{vmatrix} u_x & u_y \\ v_x & v_y
\end{vmatrix} ... |
3,100,957 | <p>A fair coin is tossed until one of the patterns show up: TTH or THT.
Let A be the event that TTH shows up before THT.</p>
<p>What is P(A)?</p>
<p>Here is my solution but I am not sure if it is correct or there is a better solution.</p>
<p>Let <span class="math-container">$p=P(A)$</span>. Define </p>
<p><span cl... | Kavi Rama Murthy | 142,385 | <p>Convergence in <span class="math-container">$\ell^{1}$</span> implies convergence of the coordinates. If a subsequence of <span class="math-container">$\{f_n\}$</span> converges, say to <span class="math-container">$f$</span>, then then we must have <span class="math-container">$f(k)=\frac 1 k$</span> for each <spa... |
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... | 5xum | 112,884 | <p>It's not yet valid, because you haven't shown why the RHS cannot be a multiple of <span class="math-container">$4$</span>. You cannot simply set <span class="math-container">$u=2$</span>, because the <span class="math-container">$u$</span> you have is already determined by <span class="math-container">$b$</span>, si... |
2,653,483 | <p>Let $a =111 \ldots 1$, where the digit $1$ appears $2018$ consecutive times.</p>
<p>Let $b = 222 \ldots 2$, where the digit $2$ appears $1009$ consecutive times.</p>
<p>Without using a calculator, evaluate $\sqrt{a − b}$.</p>
| Ng Chung Tak | 299,599 | <p>\begin{align}
a &= \frac{10^{2018}-1}{9} \\
b &= \frac{2(10^{1009}-1)}{9} \\
a-b &= \frac{10^{2018}-2\times10^{1009}+1}{9} \\
&= \frac{(10^{1009}-1)^{2}}{9} \\
\end{align}</p>
<p><strong>Can you proceed?</strong></p>
|
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>... | Randall | 464,495 | <p>Limit-compare to $\sum_n \frac{1}{\sqrt{n}}$. You need the limit
$$
\lim_{n \to \infty} \frac{\sin(1/\sqrt{n})}{1/\sqrt{n}}
$$
which is $1$ by a change of variables, making use of
$$
\lim_{t \to 0^+} \frac{\sin t}{t}=1.
$$
To get from point A to point B, change variables by $t=\frac{1}{\sqrt{n}}$. As $n \to \infty... |
52,079 | <p>I'm in doubt about the topology of maps between fibres of vector bundles.</p>
<p>Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all linear maps $T:E_x \rightarrow F_y$, where $E_x$ is the fiber over $x$ and $F_y$ is the fiber over $y$.</p>
<... | Johannes Ebert | 9,928 | <p>Let $G_i$, $i=0,1$, be topological groups and $P_i \to X_i$ be $G_i$-principal bundles. Then $P_0 \times P_1 \to X_0 \times X_1$ is a $G_0 \times G_1$-principal bundle. Let $V_i$ be topological vector spaces with continuous $G_i$-actions. The group $G_0 \times G_1$ acts on $Hom(V_0,V_1)$. If the topology on the Hom-... |
2,365,933 | <p>I'm aware of how we can simplify functions which have $Arc$ as an argument . For example $\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$ but what about cases which $Arc$ is out of the parentheses ? For instance consider this : $\sin^{-1}(\tan x)$ . Is there any way for simplification ? </p>
| Sidharth Ghoshal | 58,294 | <p>You can squeeze some more out of this.</p>
<p>Consider <span class="math-container">$$ f = \ln ( \sec x + \tan x)$$</span></p>
<p>It’s easy to see that <span class="math-container">$f’ = \sec x$</span>. Now consider <span class="math-container">$g = sin^{-1} (i \tan(x))$</span></p>
<p>We have that <span class="ma... |
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 ... | Ian Agol | 1,345 | <p>This is not what you're looking for, but I always remember Milnor's diagram in Chapter 9 of his book on <a href="https://www.maths.ed.ac.uk/%7Ev1ranick/papers/milnmors.pdf" rel="noreferrer">Morse Theory</a> describing the symmetries of the curvature tensor.</p>
<p><a href="https://i.stack.imgur.com/WwXkX.png" rel="n... |
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... | Harish Chandra Rajpoot | 210,295 | <p>Notice, $$\lim_{n\to \infty}\frac{n}{2}\sqrt{2-2\cos\left(\frac{2\pi}{n}\right)}$$
$$=\lim_{n\to \infty}\frac{n}{2}\sqrt{2\left(1-\cos\left(\frac{2\pi}{n}\right)\right)}$$ $$=\lim_{n\to \infty}\frac{n}{2}\sqrt{2\left(2\sin^2\left(\frac{\pi}{n}\right)\right)}$$ $$=\lim_{n\to \infty}\frac{2n}{2}\sin\left(\frac{\pi}{n}... |
200,903 | <p>My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was <br> <br>
Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . <br> <br></p>
<p>My teacher said that there are no solutions for this function when it is greater than $0$ and used $b^2-4ac \lt 0$, and this ... | i. m. soloveichik | 32,940 | <p>In order for $f(t)=at^2+5t+4>0$ all $t$, then the discriminant must be negative, else there are real roots to $f(t)$. The discriminant is $25-16a$, so $25-16a<0$ and thus $a>25/16$.</p>
|
103,675 | <p>I have defined a recursive sequence</p>
<pre><code>a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
</code></pre>
<p>because I want to calculate the <code>Limit</code> for this sequence when n tends towards infinity.</p>
<p>Unfortunately I get a <code>recursion exceeded</code> error when doing:</p>
<pre><code>Limit[a[n... | Daniel Lichtblau | 51 | <p>[Possibly this is too cheap and should just be a comment, I'm not sure.]</p>
<p>One way to go about such problems is to realize that "in the limit", <code>a[n]==a[n-1]</code>. So just solve an algebraic equation. If there are multiple solutions then you need to figure out which is correct based on initial values an... |
103,675 | <p>I have defined a recursive sequence</p>
<pre><code>a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
</code></pre>
<p>because I want to calculate the <code>Limit</code> for this sequence when n tends towards infinity.</p>
<p>Unfortunately I get a <code>recursion exceeded</code> error when doing:</p>
<pre><code>Limit[a[n... | J. M.'s persistent exhaustion | 50 | <p>As of version 11.2, this can be dealt with by <code>RSolveValue[]</code>:</p>
<pre><code>RSolveValue[{a[n] == Sqrt[3] + 1/2 a[n - 1], a[0] == 1}, a[∞], n]
2 Sqrt[3]
</code></pre>
|
3,497,420 | <p>Consider the function <span class="math-container">$$f(x,y)=x^6-2x^2y-x^4y+2y^2.$$</span> The point <span class="math-container">$(0,0)$</span> is a critical point. Observe,
<span class="math-container">\begin{align*}
f_x & = 6x^5-4xy-4x^3y, f_x(0,0)=0\\
f_y & = 2x^2-x^4+4y. f_y(0,0)=0\\
f_{xx} & = 30x... | Luca Goldoni Ph.D. | 264,269 | <p>You have that
<span class="math-container">$$
g_a(x)=f(x,ax^2 ) = 2\left( {a^2 - a} \right)x^4 + \left( {1 - a} \right)x^6
$$</span>
With <span class="math-container">$0<a<1$</span> the function <span class="math-container">$g_a(x)$</span> has a local maximum. With <span class="math-container">$a>1$</spa... |
3,826,994 | <p>I would like to find <span class="math-container">$z$</span> which minimizes the below, when <span class="math-container">$x$</span> is held at a specific value.</p>
<p><span class="math-container">$f(x,z) =\sqrt{\sqrt{x^2 + z^2} - 0.25}$</span></p>
<p>For example; I would like to find the value of <span class="math... | Harshit Raj | 782,789 | <p>As you said solving gets you here:
<span class="math-container">$$z = \frac{-2b +6b^2-6bi +2i}{1+b^2}$$</span></p>
<p>and finally, substituting gets u here:
<span class="math-container">$$x = \frac{-2b +6b^2}{1+b^2}$$</span> and <span class="math-container">$$y=\frac{-6b +2}{1+b^2}$$</span></p>
<p>Just divide above ... |
1,572,045 | <p>This is maybe a stupid question, but I want to find the roots of:</p>
<blockquote>
<p>$$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$$</p>
<p>So the roots are when $A$ and $B$ are both zeros when... | Laurent Duval | 257,503 | <p>First advice: when you are looking for roots of a polynomial, factorize as much as you can. You have $(x-1)^2$ and $(x-1)^3$, so you can factorize $(x-1)^2$. You have $(x+2)$ and $(x+2)^2$, so you can factorize $(x+2)$. Thus you get:
$$(x+2)(x-1)^2 (2(x-1)-3(x+2))\,. $$ So:</p>
<ol>
<li>For question 1, the ... |
38,659 | <p>I know how to use Matrix Exponentiation to solve problems having linear Recurrence relations (for example Fibonacci sequence). I would like to know, can we use it for linear recurrence in more than one variable too? For example can we use matrix exponentiation for calculating ${}_n C_r$ which follows the recurrence ... | Patrick Da Silva | 10,704 | <p>You shouldn't be so dramatically holding on to those rules ; notice that they are not called theorems, but rules ; that is because they are principles that we did not prove but stated so that students could help themselves to count stuff. I am usually comfortable with basing myself on axioms to prove theorems, but w... |
691,494 | <p>Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix </p>
<p>$\mathbf{P}= \begin{bmatrix}
\frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em]
\frac{2}{3} & 0 & \frac{1}{3} & 0\\[0.3em]
\frac{2}{3} & 0 & 0 & \frac{1}{3}\\[0.3em]
... | Gareth | 79,908 | <p>The distribution for the number of time steps to move between marked states in a discrete time Markov chain is the <a href="https://en.wikipedia.org/wiki/Discrete_phase-type_distribution" rel="noreferrer">discrete phase-type distribution</a>. You made a mistake in reorganising the row and column vectors and your tra... |
3,490,329 | <blockquote>
<p>Show that a 2-dimensional subspace of the space of <span class="math-container">$2\times2$</span> matrices contains a non-zero symmetric matrix. </p>
</blockquote>
<p>I don't know if it should be written like the addition of two symmetric and skew-symmetric matrix or there is another way to show it. ... | Josh Messing | 579,368 | <p>If <span class="math-container">$V$</span> is the linear subspace in question then we can write <span class="math-container">$V$</span> as
<span class="math-container">$$
V = \{ \lambda_1A + \lambda_2B : \lambda_1,\lambda_2 \in \mathbb{R}\}
$$</span>
for two linearly independent matrices <span class="math-container... |
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... | Rebecca J. Stones | 91,818 | <p>Let $x$ be a cut vertex and properly $2$-color the graph red and green. If we delete $x$ we get some connected components: in one of these components, let the red vertices belong to $A$ and green vertices belong to $B$.</p>
<p>The graph induced by $\{x\} \cup A \cup B$ looks as follows:</p>
<p><img src="https://i... |
2,996,920 | <p>Please recommend me a good book to study interpolation techniques such as polynomial interpolation, cubic, spline interpolations, if possible tell me the branch of mathematics that deals with this subject. I want to go in depth with this topic.</p>
| TurlocTheRed | 397,318 | <p>Numerical Recipes is a highly recommended classic. I personally know working physicists who use it whenever they need some numerical work done:</p>
<p><a href="http://numerical.recipes/oldverswitcher.html" rel="nofollow noreferrer">http://numerical.recipes/oldverswitcher.html</a></p>
|
211,427 | <p>Can't seem to figure this one out. Could anyone help me out and explain it to me?<br>
Thank you.</p>
<p>Let $P$ and $Q$ be relations on $Z$ by x$P$y iff x + 1 <= y and a$Q$b iff a + 2 <= b. Prove that P $\circ$ Q = {(p,q) belonging to ZxZ | p + 3 <= q} </p>
| Michael Albanese | 39,599 | <p>I will outline how to go from a symmetric bilinear form to a linear operator.</p>
<p>Given a symmetric bilinear form $\sigma : V \times V \to F$, consider the map $L' : V \to V^*$ given by $L'(v) = \sigma(v, \cdot)$. It is not a linear operator as the codomain is $V^*$ not $V$. However, as $V$ is finite dimensional... |
3,411,081 | <p>A group of 12 people are going out to a concert on Saturday night. The group will take three cars with four people in each car. If they distribute themselves at random, what is the probability that A and B will be in the same car?</p>
<p>I tried (12C2*10C2*8C4*4C4)/(12C4*8C4*4C4) because you're choosing two first a... | Alexander Geldhof | 560,477 | <p>Let the number be <span class="math-container">$z + 1$</span>. Jot down <span class="math-container">$1$</span> on your paper.</p>
<p>Now decide whether you want to add <span class="math-container">$1$</span> to the <em>last</em> number you already wrote down (i.e. make it <span class="math-container">$2$</span>) o... |
3,411,081 | <p>A group of 12 people are going out to a concert on Saturday night. The group will take three cars with four people in each car. If they distribute themselves at random, what is the probability that A and B will be in the same car?</p>
<p>I tried (12C2*10C2*8C4*4C4)/(12C4*8C4*4C4) because you're choosing two first a... | Community | -1 | <p>Let <span class="math-container">$f(n)$</span> be the number you're looking for for a given <span class="math-container">$n$</span>. Let's work by strong induction.</p>
<p>Consider <span class="math-container">$f(n+1)$</span>. It could either have one term (which happens in one way) or more than one term. In thi... |
1,270,042 | <p>$$(a+5)(b-1)=ab-a+5b-5=20-5=15.$$</p>
<p>So, both $a + 5$ and $b-1$ divide $15$. </p>
<p>Then, $a + 5$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $a$ is one of $10, -20, -2, -8, 0, -10, -4, -6$ and $b – 1$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $b = 14, -14, 4, -2, 6, -4, 2, 0$.</p>
<p>Could all possibili... | DeepSea | 101,504 | <p>Or write $b = \dfrac{a+20}{a+5} = 1 + \dfrac{15}{a+5}$, thus $a+5 \mid 15$, and $a+5 = 1, 3, 5, 15$ are the possibilities. But since we consider only positive values, we have:$a+5 = 15 \to a = 10 \to b = 1+1 = 2$.</p>
|
2,135,191 | <p>In the sequence $a_{1}, a_{2}, a_{3}, ..., a_{100}$, the $k$th term is defined by $$a_{k} = \frac{1}{k} - \frac{1}{k+1}$$ for all integers $k$ from $1$ through $100$. What is the sum of $100$ terms of this sequence? </p>
<p>The answer given is $\frac{100}{101}$, but I am not sure how.</p>
<p>So far I am have plugg... | Arnaldo | 391,612 | <p>$$a_1+a_2+a_3+...+a_{100}=\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+...+\left(\frac{1}{99}-\frac{1}{100}\right)+\left(\frac{1}{100}-\frac{1}{101}\right)=1-\frac{1}{101}=\frac{100}{101}$$</p>
<p>Note that $-\frac{1}{2}$ cancel $\frac{1}{2}$, $-\fra... |
2,135,191 | <p>In the sequence $a_{1}, a_{2}, a_{3}, ..., a_{100}$, the $k$th term is defined by $$a_{k} = \frac{1}{k} - \frac{1}{k+1}$$ for all integers $k$ from $1$ through $100$. What is the sum of $100$ terms of this sequence? </p>
<p>The answer given is $\frac{100}{101}$, but I am not sure how.</p>
<p>So far I am have plugg... | Mark Fischler | 150,362 | <p>$$a_{100} = \left( \frac11 - \frac12 \right)+\left( \frac12 - \frac13 \right)+
\left( \frac13 - \frac14 \right)+\cdots + \left( \frac1{100} - \frac1{101} \right)
\\= \frac11 + \left( -\frac12+\frac12\right)+\left( -\frac13+\frac13\right)
+\cdots +\left( -\frac1{100}+\frac1{100}\right) -\frac1{101} = 1 - \frac1{101}
... |
635,077 | <p>$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$</p>
<p>How can I prove this statement?</p>
| DanielWainfleet | 254,665 | <p>Here is a modern proof using sleight of hand.(1)An isometry of the Euclidean plane $E$ is a function $f:E\to E$ that preserves the distance $d$ between points: $d(A,B)=d(f(A),f(B))$ for all $A,B\in E$. We have (2):An isometry $f$ is onto.Because, for $Q\in E$, let $A_1,A_2,A_3\in E$ be non-co-linear with $Q\not \in ... |
226,551 | <p><strong>(1)</strong> Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?</p>
<p><strong>(2)</strong> Is there a repository of adjacencies from such classes?</p>
| John Machacek | 51,668 | <p>In <a href="http://people.mpi-inf.mpg.de/~pascal/docs/KutzSchw_ScrewBox.pdf" rel="noreferrer">ScrewBox: a Randomized Certifying Graph-Non-Isomorphism Algorithm</a> by Kutz and Schweitzer it is stated that "It is generally accepted that the incidence graphs of finite projective planes confront graph isomorphism algor... |
226,551 | <p><strong>(1)</strong> Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?</p>
<p><strong>(2)</strong> Is there a repository of adjacencies from such classes?</p>
| Brendan McKay | 9,025 | <p>There is a move towards creating a "standard" benchmark set for graph isomorphism, but it didn't happen yet. Meanwhile, the largest collection that includes hard graphs as well as easy graphs is <a href="http://pallini.di.uniroma1.it/Graphs.html" rel="noreferrer">here</a>. To make examples of difficult nonisomorphi... |
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... | John Hughes | 114,036 | <p>Write $\ln 2$ as a binary number with bits $b_1, b_2, \ldots$. </p>
<p>As others have suggested in briefly-present answers, taking logs is the secret: you want
$$
\lim_{n\to+\infty}\frac{a_n}{2^n}=\ln(2)
$$</p>
<p>and hence
$$a_n=2^n\ln(2).$$
but this choice of $a_n$ is not an integer. On the other hand, the
f... |
886,070 | <p>This is a follow-up question to <a href="https://math.stackexchange.com/questions/884642/an-equation-of-the-form-a-b-c-abc">An equation of the form A + B + C = ABC</a> . I totally messed up with making the equation from the question specification . Actually the question was $$ \arctan(\frac{1}{A}) = \arctan(\frac{1... | Robert Israel | 8,508 | <p>Write your equation as $$B = \dfrac{AC+1}{C-A}$$
Let $C = A + x$, so this becomes
$$ B =A + \dfrac{A^2+1}{x}$$
Thus $x$ must divide $A^2+1$, and
$B + C = 2A + x + \dfrac{A^2+1}{x}$.
You'll want a divisor of $A^2+1$ that is closest to $\sqrt{A^2+1}$ (on one side or the other).</p>
|
886,070 | <p>This is a follow-up question to <a href="https://math.stackexchange.com/questions/884642/an-equation-of-the-form-a-b-c-abc">An equation of the form A + B + C = ABC</a> . I totally messed up with making the equation from the question specification . Actually the question was $$ \arctan(\frac{1}{A}) = \arctan(\frac{1... | The Great Seo | 166,806 | <p>$$\begin{align}
\frac1A={B+C\over BC-1} &\iff BC-AB-AC-1=0 \\&\iff (B-A)(C-A)=BC-BA-BC+A^2=A^2+1.
\end{align}$$
Thus as Robert Isreal pointed out, to minimize $B+C$, you have to find a divisor of $A^2+1$ that is closest to $\sqrt{A^2+1}$, and let it be $B-A$.</p>
|
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>
| Barry Cipra | 86,747 | <p>Let $f(x)=1-\left({3\over4}\right)^x-\left({1\over4}\right)^x$. The functions $(3/4)^x$ and $(1/4)^x$ are strictly decreasing functions of $x$, so the minus signs make $f(x)$ strictly increasing. </p>
|
279,520 | <p>So I have to find the integral of $$ \int \frac{\sin^{-1}(x)}{\sqrt{1+x}} \; dx$$</p>
<p>I think I have to do this using the integration by parts..so I will take $f = \sin^{-1}(x)$ and $ \sqrt {1+x}=g' $...what about now? </p>
| Stefan Hansen | 25,632 | <p>Do integration by parts with $f(x)=\arcsin(x)$ and $g(x)=\frac{1}{\sqrt{1+x}}$. Then
$$
I=\int\frac{\arcsin x}{\sqrt{1+x}}\,\mathrm dx=\int f(x)g(x)\,\mathrm dx=f(x)G(x)-\int f'(x)G(x)\,\mathrm dx\\
=2\arcsin (x)\sqrt{1+x}-\int \frac{1}{\sqrt{1-x^2}}\cdot2\sqrt{1+x}\,\mathrm dx,
$$
but
$$
\frac{1}{\sqrt{1-x^2}}\cdot... |
3,057,198 | <p><span class="math-container">$$\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... | Greg Martin | 16,078 | <p>In double integrals of the form <span class="math-container">$\int (\int f(x,y)\,dx)\,dy$</span>: the limits of the outer integral are the largest possible values of <span class="math-container">$y$</span> over the entire domain of integration; then, for every fixed <span class="math-container">$y$</span>, the limit... |
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}]... | coudy | 716,791 | <p>The <a href="https://en.wikipedia.org/wiki/Quartic_function" rel="nofollow noreferrer">formulas</a> solving the quartic involve taking cube roots in general. Some <a href="https://en.wikipedia.org/wiki/Resolvent_cubic" rel="nofollow noreferrer">auxiliary cubic</a> equation is needed in the resolution of a quartic in... |
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... | Tyma Gaidash | 905,886 | <p>Unfortunately, there is no closed form so far, at least without using special functions, but here it is using this <a href="https://www.desmos.com/calculator/eaqmpcq4t7" rel="nofollow noreferrer">graph</a>. You can also think of plugging in some value of x into the final answer at the top of the power tower like ans... |
3,371,922 | <p>The definition of the limit states that limit of <span class="math-container">$f(x)$</span> when <span class="math-container">$x$</span> approaches <span class="math-container">$c$</span> is <span class="math-container">$L$</span> iff for every <span class="math-container">$\epsilon > 0$</span> there exists <spa... | Theo Bendit | 248,286 | <p>The definition is not quite what you wrote. It should be:</p>
<blockquote>
<p>For all <span class="math-container">$\varepsilon > 0$</span>, there exists <span class="math-container">$\delta > 0$</span> such that, <strong>if</strong> <span class="math-container">$0 < |x - c| < \delta$</span>, <strong>... |
106,126 | <blockquote>
<p><strong>Problem</strong> Prove that $n! > \sqrt{n^n}, n \geq 3$. </p>
</blockquote>
<p>I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder ... | André Nicolas | 6,312 | <p>There can be no method prettier than Henry's pairing argument. So let's work in the opposite direction, and look for an ugly argument.</p>
<p>When we go from $n$ to $n+1$, the factorial function grows by a factor of $n+1$. What about the other function? It grows by the factor
$$\frac{\sqrt{(n+1)^{n+1}}}{\sqrt{n^n}... |
106,126 | <blockquote>
<p><strong>Problem</strong> Prove that $n! > \sqrt{n^n}, n \geq 3$. </p>
</blockquote>
<p>I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder ... | robjohn | 13,854 | <p>$\log$ is strictly concave $\left(\frac{\mathrm{d}^2}{\mathrm{d}x^2}\log(x)=-\frac{1}{x^2}<0\right)$, so for $1\le k\le n$, we have
$$
\log(k)\ge\frac{(k-1)\log(n)+(n-k)\log(1)}{n-1}\tag{1}
$$
with equality only when $k=1$ or $k=n$. Summing $(1)$ yields
$$
\begin{align}
\log(n!)
&\ge\frac{n}{2}(\log(n)+\... |
222,596 | <p>I would like to find a temperature by knowing the enthalpy, is this possible?
This is what i've tried so far:</p>
<pre><code>V1 = 150;
V2 = 4;
T1 = 15 + 273;
Enthalpy
h[T_] := QuantityMagnitude[
ThermodynamicData["Air",
"Enthalpy", {"Temperature" -> Quantity[T, "Kelvins"]}]]
h1 = h[T1]
sol = ... | Tim Laska | 61,809 | <p>I would explicitly state the reference pressure (assuming 1 Bar). Here is an alternative way:</p>
<pre><code>Clear[h]
h[t_?NumericQ] :=
QuantityMagnitude@
ThermodynamicData["Air",
"Enthalpy", {"Pressure" -> Quantity[1, "Bars"],
"Temperature" -> Quantity[t, "Kelvins"]}]
V1 = 150;
V2 = 4;
T1 = 15 ... |
787,926 | <p>I need some help to solve this integral:</p>
<p>$$\int_0^1 dy\int_0^{1-y} \cos \left(\frac{x-y}{x+y} \right) \mathrm dx$$</p>
<p>Thank you.</p>
| 2'5 9'2 | 11,123 | <p>Every $5$-smooth number is of the form $2^x3^y5^z$. Let's call the $1000$th one $M$. That means there are $1000$ nonnegative integer solutions to $$2^x3^y5^z<M$$ This is the same as having $1000$ nonnegative integer solutions to $$\log(2)x+\log(3)y+\log(5)z<\log(M)$$</p>
<p>There are approximately as many non... |
4,177,829 | <p>Given angles <span class="math-container">$0<\theta_{ij}<\pi$</span> for <span class="math-container">$1\leq i<j\leq k$</span>, what conditions are there on the angles to ensure that there exists <span class="math-container">$k$</span> unit vector <span class="math-container">$v_i\in \mathbb R^k$</span> so ... | Ѕᴀᴀᴅ | 302,797 | <p><span class="math-container">$\def\vec{\boldsymbol}\def\v{\vec{v}}\def\x{\vec{x}}\def\R{\mathbb{R}}\def\Ω{{\mit Ω}}\def\<{\langle}\def\>{\rangle}\DeclareMathOperator{\diag}{diag}$</span>On the one hand, if there exists such <span class="math-container">$\v_1, \cdots, \v_n$</span>, define <span class="math-cont... |
3,045,899 | <p>In the lecture notes we have a fact:</p>
<blockquote>
<p>If <span class="math-container">$A$</span> has orthonormal columns then <span class="math-container">$||Ax||^2_2 = ||x||^2_2$</span></p>
</blockquote>
<p>Why is it the case? What properties of matrix-vector multiplication should I know to reason about this... | C. Ding | 320,080 | <p>Since <span class="math-container">$A$</span> has orthonormal columns, <span class="math-container">$A^TA=I$</span>.</p>
|
4,407,210 | <p><span class="math-container">$$y''-\frac{1}{x}y'=2x\cdot cos(x)$$</span></p>
<p>For the homogeneous part I multiplied through with <span class="math-container">$x^2$</span> and got a second order Cauchy Euler equation with the general solution: <span class="math-container">$$y_h (x)=A x^2 +B$$</span></p>
<p>Then for... | user577215664 | 475,762 | <p><span class="math-container">$$y''-\frac{1}{x}y'=2x\cos(x)$$</span>
<span class="math-container">$$\dfrac {xy''-y'}{x^2}=2\cos(x)$$</span>
<span class="math-container">$$\left (\dfrac {y'}{x} \right)'=2\cos(x)$$</span>
Integrate.</p>
|
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_... | Michael Greinecker | 21,674 | <p>This is a consequence of Ulam's theorem: If a finite countable additive measure $\mu$ is defined on all subsets of a set of cardinality $\aleph_1$ (the first uncountable cardinal) and vanishes on all singletons, then it is identically zero.</p>
<p>This is the version of the theorem from the book of <a href="http://... |
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>
| Michael Hardy | 11,667 | <p>$3\times8$ means $8+8+8$.</p>
<p>$8\times3$ means $3+3+3+3+3+3+3+3$.</p>
<p>Why should those both be the same number?</p>
<p>Why should $a\times b$ always be the same as $b\times a$?</p>
|
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>
| Jyrki Lahtonen | 11,619 | <p>I am bit undecided about the following riddles, whether they are too difficult for 9-year-olds. Here comes anyway. </p>
<ol>
<li>A boy was selling eggs (replace eggs with whatever works locally) to people in a building with four floors. The person living in the fourth floor bought half the eggs plus half an egg. Th... |
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>
| Naffi | 82,231 | <p>Show them a pizza. Ask how many pieces they can make by cutting it once.
<img src="https://i.stack.imgur.com/Penoh.jpg" alt="enter image description here"></p>
<p>The answer should be 2. Now ask them for 2 cutting. The answer should be 4. Then ask them to try for 3 cuttings. I hope some of them will the correct ans... |
4,206,286 | <p><span class="math-container">$$\int_0^1\int_0^\infty ye^{-xy}\sin x\,dx\,dy$$</span></p>
<p>How can I calculate out the value of this integral?</p>
<p>P.S. One easy way is to calculate this integral over <span class="math-container">$dy$</span> first, to get an integral form <span class="math-container">$\frac{1-e^{... | sirous | 346,566 | <p><a href="https://i.stack.imgur.com/t4Q8T.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/t4Q8T.jpg" alt="" /></a></p>
<p>Hints:</p>
<p>For equilateral triangle shown in figure a it is obvious:</p>
<p><span class="math-container">$$(d_c=r)^2=(OB=R)^2-\big(\frac{AB=c}2\big)$$</span></p>
<p><span cla... |
3,779,589 | <p>Let the metric <span class="math-container">$d$</span> be defined as
<span class="math-container">$$
d(f,g) =\sup_{x\in[0,1]}|f(x)-g(x)|,
$$</span>
and let<br />
<span class="math-container">$$
H(x) = \begin{cases} 0 \text{ if } x \leq \frac{1}{2}\\ 1 \text { if } x > \frac{1}{2} \end{cases}.
$$</span>
Is <span c... | user2661923 | 464,411 | <p>I agree with the other responses but offer an <strong>informal</strong> [i.e. intuitive rather than proven] alternative approach.</p>
<p>Consider the numbers <span class="math-container">$1, 2, \cdots, 10.$</span> Of these 10 numbers, exactly 4 are not divisible by either 2 or 5. Further, <span class="math-containe... |
1,005,193 | <p>My problem is as follows:</p>
<p>I have a point $A$ and a circle with center $B$ and radius $R$. Points $A$ and $B$ are fixed, also $A$ is outside of the circle. A random point $C$ is picked with uniform distribution in the area of disk $B$. My question is how to calculate the expected value of $AC^{-4}$. I am work... | Vladimir Vargas | 187,578 | <p>Notice that:</p>
<p>$$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).$$</p>
|
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| JSE | 431 | <p>Slowly and with difficulty, just like amateur mathematicians.</p>
|
240,461 | <p>What's the mathematica command to get the <strong>numerical value</strong> of :</p>
<p><span class="math-container">$$PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$$</span></p>
<p>where <span class="math-container">$PV$</span> is the principal value.</p>
| Dominic | 47,466 | <p>You can evaluate principal-valued integrals using NIntegrate using Method->"PrincipalValue" but you must specify the singular point(s). In your case, there are an infinite number of them. For starters, integrating over just the first one at <span class="math-container">$\pi/2$</span>, we can write:</p... |
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