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 |
|---|---|---|---|---|
3,819,639 | <p>Integral: <span class="math-container">$J=\int_0^1 \frac{x}{1+x^8}dx$</span></p>
<p>Consider the following assertions:</p>
<p><span class="math-container">$I:J> \frac{1}{4}$</span> and <span class="math-container">$II:J< \frac{\pi}{8}$</span></p>
<p>A. Both are true</p>
<p>B. Only <span class="math-container">... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,799,710 | <p>The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has</p>
<p>$1.$ Only one particular integral.</p>
<p>$2.$ a particular integral which is linear in x and .</p>
<p>$3.$a particular integral which is a quadratic polynomial is x and ... | John Cramerus | 334,508 | <p>So first things first, your answer is correct. Since you're confused, it means that you don't totally understand the process that you used to get to your answer, so let's break it down step by step.</p>
<p>The first thing you did is put each new basis vector in $B' = \{v_1, v_2, v_3\}$ in terms of your original bas... |
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | Timothy Wagner | 3,431 | <p>Assume, <span class="math-container">$n\geq 4$</span>. By Bertrand's postulate there is a prime, let's call it <span class="math-container">$p$</span> such that <span class="math-container">$\frac{n}{2}<p<n$</span> . Suppose, <span class="math-container">$p^2$</span> divides <span class="math-container">$n$</s... |
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| G Tony Jacobs | 92,129 | <p>Just look at a graph of $y=1/t$ between $t=n-1$ and $t=n$, versus the constant function $y=1/n$ between the same two $t$-values. The latter graph is a rectangle, with its area contained entirely under the first graph.</p>
|
484,273 | <p>$$\int_0^1\frac{ \arcsin x}{x}\,\mathrm dx$$</p>
<p>I was looking in my calculus text by chance when I saw this example , the solution is written also but it uses very tricky methods for me ! I wonder If there is a nice way to find this integral.</p>
<p>The idea of the solution in the text is in brief , Assume $y... | Argon | 27,624 | <p>$$I = \int_0^1 \frac{\arcsin x}{x}\,dx = \int_{0}^{\pi/2}\frac{x\,dx}{\sin x}\cos x = x\log\sin x \bigg|_0^{\pi/2}-\int_0^{\pi/2} \log\sin x\,dx$$</p>
<p>This last integral is <a href="https://math.stackexchange.com/questions/37829/computing-the-integral-of-log-sin-x">well known</a> to equal $-\frac{\pi}{2}\log 2$.... |
2,001,449 | <p>Where in the analytic hierarchy is the theory of all true sentences in ZFC? In higher-order ZFC? In ZFC plus large cardinal axioms?</p>
<p>Edit: It seems that this is ill-defined. Why is this ill-defined for ZFC, but true for weaker theories like Peano arithmetic and higher-order arithmetic?</p>
| goblin GONE | 42,339 | <p>Truth has to be defined relative to an intended interpretation. Unfortunately, ZFC doesn't have one; in particular, the usual story that ZFC is intended as a theory of the von Neumann universe fails to tell us which large cardinals exist. So in some sense, we don't really know what ZFC is supposed to "mean".</p>
<p... |
3,369,669 | <blockquote>
<p>let <span class="math-container">$V$</span> be a real finite dimensional vector space and <span class="math-container">$f,g$</span> are nonzero linear functional on <span class="math-container">$V$</span> real vector space,Assume that <span class="math-container">$Ker(f)\subset Ker(g)$</span> ... | Chinnapparaj R | 378,881 | <p>Note that <a href="https://math.stackexchange.com/questions/1827627/show-that-ker-t-is-a-maximal-linear-subspace-of-v">null spaces of linear functionals are maximal subspaces</a>. So <span class="math-container">$\ker f$</span> and <span class="math-container">$\ker g$</span> are maximal subspaces and <span class="... |
5,890 | <p>Is there a way to <em>temporarily</em> suppress certain messages, so that I could write for example (with made-up syntax for that feature):</p>
<pre><code>WithOff[Pattern::patv, rule = (f[x_Integer|{x__Integer}] :> g[x])];
rule2 = x_[x__] :> x;
</code></pre>
<p>and get no <code>Pattern::patv</code> message f... | Peter Breitfeld | 307 | <p>You can define the function:</p>
<pre><code> messageIsOn[msg_]:=Head[msg]===String
</code></pre>
<p>Which yields True if the message is on. Then do e.g:</p>
<pre><code>msgStatus=messageIsOn[Pattern::patv]
If[msgStatus, Off[Pattern::patv]]
< some calculation suppressing message Pattern::patv >
(* Restore the... |
1,400,109 | <p>I am wondering if anyone can help me find an analytical solution to the roots of the following function:
<span class="math-container">$$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$</span>
<span class="math-container">$a,b \in (0,1)$</span> and <span class="math-container">$n... | Claude Leibovici | 82,404 | <p>I am not sure that you can have an explicit solution and I am afraid that numerical methods should be required.</p>
<p>We have $$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right)$$ $$f'(b)=\frac{c-b n}{b(1-b)}$$ $$f''(b)=-\frac{nb^2 -2 cb +c}{(b-1)^2 b^2}$$ Since $c<n$, $f''(b)$... |
24,524 | <p>Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular faces'). Is it true than one can extend to a morphism of such diagrams:
1. any morphism of one of the 'commutative faces... | S. Carnahan | 121 | <p>(Note: This does not answer your question. <b>Edit: </b> It still doesn't answer your question, but I am attempting to make my notation less horribly broken, so future readers don't get too confused.)</p>
<p>I'm not entirely sure which parts of the diagram you are considering, so I'll try to fix some notation. We ... |
2,929,238 | <p>Recently I have come arcross the following fraction</p>
<blockquote>
<p><span class="math-container">$$\dfrac{z^2}{(2z^2+3)((s^2+1)z^2+1)}$$</span></p>
</blockquote>
<p>Hence I have encountered this fraction within a task of integration I want to do a partial decomposition. First of all I rewrote it as following... | Acccumulation | 476,070 | <p>Given any unit vectors, there is a nonsingular rotation matrix that takes the first vector to the other. Given any non-zero, non-unit vector, there is some non-singular scaling matrix that takes the vector to a unit vector. So given two arbitrary non-zero vectors <span class="math-container">$v$</span> and <span cla... |
3,057,874 | <blockquote>
<p>The following formula shall be proved by induction:
<span class="math-container">$$F(m+n) = F(m-1) \cdot F(n) + F(m) \cdot F(n+1)$$</span>
Where <span class="math-container">$F(i), i \in \mathbb{N}_0$</span> is the Fibonacci sequence defined as:
<span class="math-container">$F(0) = 0$</span>, <s... | J.G. | 56,861 | <p>I won't mention every use of induction. Define <span class="math-container">$$M:=\left(\begin{array}{cc}
0 & 1\\
1 & 1
\end{array}\right),\,V_{n}:=\left(\begin{array}{c}
F_{n}\\
F_{n+1}
\end{array}\right)=M^{n}\left(\begin{array}{c}
0\\
1
\end{array}\right)$$</span> so <span class="math-container">$M^{m}=\le... |
4,650,606 | <p>Let <span class="math-container">$X$</span> be a complex Banach space, and let <span class="math-container">$P$</span> be a bounded linear operator acting on the dual <span class="math-container">$X^{*}$</span> such that that <span class="math-container">$P^2=P$</span>.
I research for a bounded linear operator <span... | Ryszard Szwarc | 715,896 | <p>Let <span class="math-container">$\varphi $</span> denote the invariant mean on <span class="math-container">$\ell^\infty(\mathbb{N})$</span> (or any continuous extension of the limit on the space of convergent sequences). Define <span class="math-container">$$Px^*=\varphi(x^*)\mathbf{1},\quad x^*\in \ell^\infty$$</... |
2,080,460 | <p>The set of uniqueness for $H^2$ is defined to be a set $E\subseteq \mathbb{D}$ such that if $f\in H^2$ and $f|_E =0$ then $f\equiv 0$. Let $$ k_\lambda(z) =\frac{1}{1-z\ \overline{\lambda}}$$ be the reproducing kernel for $H^2$. I want to prove a statement that is claimed in a text and the statement is : If $\{\lamb... | Dietrich Burde | 83,966 | <p>Actually there is a generalization of the Mordell-Weil Theorem, which is as follows:</p>
<p><strong>Theorem</strong> (Mordell-Weil-Lang-Neron) <em>Let $K$ be a field that is of finite type over its prime field (where the prime field is either $\mathbb Q$ or $\mathbb F_p$), and let $A/K$ be an abelian variety. Then ... |
2,181,989 | <p>In all the question I never took $\sin x =t$ and was able to solve most of them but I got stack in this question and I saw the solution , I found that they took $\sin x =t$ and treated $\sin x$ as just a variable $t$. I found it a bit weird, now many question how would I had known that I had to take $\sin x$ as $t$ ... | egreg | 62,967 | <p>The idea is that, in a neighborhood of $0$, the function $\sin x$ is continuous and invertible, so also its inverse is likewise continuous.</p>
<p>You can consider the function
$$
f(t)=\frac{t+\log(\sqrt{1+t^2}-t)}{t^3}
$$
and you have to compute
$$
\lim_{x\to0}f(\sin x)
$$
Because of the above fact, this is the sa... |
101,838 | <p>I am trying to factor the following polynomial: $$ 8x^3 -4x^2y -18xy^2 + 9y^3 $$</p>
<p>$$ (a-b)^3 = a^3 -3a^2b + 3ab^2 - b^3 $$
Thanks</p>
| André Nicolas | 6,312 | <p>Hint: Maybe look at the pretty much equivalent problem of factoring
$8x^3-4x^2-18x+9$.</p>
<p>We can use the Rational Roots Theorem to find the rational roots of this, if any (and there are). We can also make life simpler by writing $2x=w$, which yields
$w^3-w^2-9w+9$. </p>
<p>Or else we can note that
$8x^2-4x^2-... |
2,138,663 | <p>Suppose a function $f$ is Riemann integrable over any interval $[0,b]$. By definition the improper integral is convergent if there is a real number $I$ such that</p>
<p>$$\lim_{b \to \infty}\int_0^b f(x) dx= I := \int_0^\infty f(x)dx.$$</p>
<p>I have shown that if $f$ is nonnegative then this is equivalent for $n ... | RRL | 148,510 | <p>If the improper integral converges conditionally and not absolutely, then the limit of $\int_{A_n} f$ need not exist. This is somewhat surprising since $A_n \subset A_{n+1}$ and $A_n \uparrow [0,\infty).$ </p>
<p>For example, it is well known that the improper integral of $f(x) = \sin x / x$ converges conditi... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Community | -1 | <p>Assume that the formula $\sin'_r(x)=\cos_r(x)$ is true for some angular unit, let "$r$". For another angular unit, let "$d$", there is a conversion factor, let $\lambda_{d\rightarrow r}$, and we can write:</p>
<p>$$\sin_d'(x)=\sin_r'(\lambda_{d\rightarrow r} x)=\lambda_{d\rightarrow r}\cos_r(\lambda_{d\rightarrow r... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| user132181 | 132,181 | <p>Simple answer: radian is not really a unit, it's an absence of one. Degree, on the other hand, is. Working with dimensionful quantities in calculus is the last thing you would want to do (unless you're into kinky things) :)</p>
|
2,092,814 | <p>How to prove the following series,
$$\sum_{k=1}^{\infty }\frac{\left ( -1 \right )^{k}}{k}\sum_{j=1}^{2k}\frac{\left ( -1 \right )^{j}}{j}=\frac{\pi ^{2}}{48}+\frac{1}{4}\ln^22$$
I know a formula which might be usful.
$$\sum_{j=1}^{n}\frac{\left ( -1 \right )^{j-1}}{j}=\ln 2+\left ( -1 \right )^{n-1}\int_{0}^{1}\fra... | Renascence_5. | 286,825 | <p>Yes! As you mentioned, it's a very useful formula.</p>
<p>Use
$$\sum_{j=1}^{n}\frac{\left ( -1 \right )^{j-1}}{j}=\ln 2+\left ( -1 \right )^{n-1}\int_{0}^{1}\frac{x^{n}}{1+x}\, \mathrm{d}x$$
we get
\begin{align*}
&\sum_{k=1}^{\infty }\frac{\left ( -1 \right )^{k}}{k}\sum_{j=1}^{2k}\frac{\left ( -1 \right )^{j}... |
3,024,496 | <p>I have this determinant which looks like a Vandermonde matrix</p>
<p><span class="math-container">$$D=\begin{vmatrix}1& a_1 & \cdots & a_1^{n-2}& a_1^n\\
1& a_2 & \cdots & a_2^{n-2}& a_2^n\\
\vdots &\vdots & \ddots & \vdots & \vdots\\
1& a_n & \cdots & a_n... | Mike Earnest | 177,399 | <p>You formula is correct. My proof is not pretty.</p>
<p>To compute <span class="math-container">$D$</span>, subtract the last row of the matrix from each of the other rows. The <span class="math-container">$i^{th}$</span> row will now have a factor of <span class="math-container">$a_i-a_n$</span>, for all <span clas... |
3,024,496 | <p>I have this determinant which looks like a Vandermonde matrix</p>
<p><span class="math-container">$$D=\begin{vmatrix}1& a_1 & \cdots & a_1^{n-2}& a_1^n\\
1& a_2 & \cdots & a_2^{n-2}& a_2^n\\
\vdots &\vdots & \ddots & \vdots & \vdots\\
1& a_n & \cdots & a_n... | jjagmath | 571,433 | <p>Hint:</p>
<p>Consider <span class="math-container">$$D =
\begin{vmatrix}
1&x&x^2&\cdots&x^{n-2} & x^{n-1}&x^n\\
1&a_1&a_1^2&\cdots&a_1^{n-2} & a_1^{n-1}&a_1^n\\
1&a_2&a_2^2&\cdots&a_2^{n-2} & a_2^{n-1}&a_2^n\\
\vdots\\
1&a_n&a_n^2&... |
2,766,332 | <blockquote>
<p>If $a,b$ are elements of a group and $a^2=e, b^6=e, ab=b^4a$, then find the order of $ab$ and express ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$ </p>
</blockquote>
<p>I could find the order of $ab$ to be 6 but struggling to find ${(ab)}^{-1}$ in terms of $a^mb^n$ and $b^ma^n$.</p>
<p>Please hel... | Joffan | 206,402 | <p>With the information given, you could actually simplify the powers of $ab$ in succession</p>
<p>$(ab)^2 = b^4a\cdot ab = b^5$<br>
$(ab)^3 = ab\cdot b^5 = a$<br>
$(ab)^4 = a\cdot ab = b$<br>
$(ab)^5 = ab\cdot b = ab^2$<br>
Also $(ab)^5 = b\cdot b^4a = b^5a$<br>
And finally $(ab)^6 = b^5a\cdot ab = b^6=e$ meaning th... |
1,048,045 | <blockquote>
<p>$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$</p>
</blockquote>
<p>I try to solve it, but failed. Who can help me to find it?</p>
<p>I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\cos(x)}{1+\sin^2(x)}\,dx}$.</p>
| Aditya Hase | 190,645 | <blockquote>
<p>Actually I didn't post my answer as I was unable to continue it where I left but after Pranav provided link I'm posting it</p>
</blockquote>
<p>$$\sin x=u\iff \frac{\,\mathrm du}{\sqrt{1-u^2}}=\,\mathrm dx$$</p>
<p>$$\int_0^{{\pi/2}}\arctan\left(\sin x\right)\,\mathrm dx=\int_0^{1}\frac{\arctan(u)}{... |
159,789 | <p>$$\lim_{r \to 0^+} \frac{\sqrt r}{(r-9)^4}\
$$</p>
<p>How do i compute this limit? I was told to see what 1/x is approaching and rewrite it but can someone guide me in the right direction?</p>
<p>How can i find which infinity it is approaching?</p>
<p>Also</p>
<p>What does it approach if the limit approach 9 i... | André Nicolas | 6,312 | <p>We need to look at what happens to top and to bottom as $r$ approaches $0$ from the right. The bottom behaves very nicely: as $r$ approaches $0$, $(r-9)^4$ approaches $(-9)^4$.</p>
<p>The top also behaves nicely: as $r$ approaches $0$ from the right, $\sqrt{r}$ approaches $0$.</p>
<p>So the quotient approaches $0/... |
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | Peter Gerdes | 23,648 | <p>I'm 100% in agreement about the need for such a site. In fact I'm currently in the process of coding what I hope can grow into such a site.</p>
<p>The apps mentioned above like Papers<span class="math-container">$^\gamma$</span> are all nice but they all just kinda throw the user at a list of papers and say: go com... |
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | mathandy | 48,532 | <p>I'm one of the founders of <a href="http://info.inquire.pub" rel="noreferrer">info.inquire.pub</a> (main site not yet available to the public). We're a non-profit startup with a similar purpose. We aim to be a Q&A site with a reputation system and upvotes, where each question is associated with a specific paper... |
1,971,645 | <p>I tried doing this problem two ways. I am unable to get the solutions to match each other. Is one of them incorrect?</p>
<p><a href="https://i.stack.imgur.com/5vHwDxx.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5vHwDxx.jpg" alt="enter image description here"></a></p>
| Adi Dani | 12,848 | <p>$$y+\sqrt{xy}=1$$
$$y+\sqrt{x}\sqrt{y}=1$$
$$y'+\frac{1}{2\sqrt{x}}\sqrt{y}+\frac{y'}{2\sqrt{y}}\sqrt{x}=0$$
$$y'(1+\frac{\sqrt{x}}{2\sqrt{y}})+\frac{1}{2\sqrt{x}}\sqrt{y}=0$$
$$y'=\frac{-\frac{1}{2\sqrt{x}}\sqrt{y}}{1+\frac{\sqrt{x}}{2\sqrt{y}}}=\frac{-y}{2\sqrt{xy}+x}=\frac{y}{2y-2-x}$$</p>
|
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | Dr. Michael W. Ecker | 438,184 | <p>We have two circles with different centers, which centers we align along the x-axis in the x, y plane. In particular, we place the origin of the x, y plane at the center of one circle. That circle has equation x^2 + y^2 = r^2, while the other has equation (x-h)^2 + y^2 = R^2, for some h > 0. Thus, if the circles ... |
2,853,673 | <p>I came across this as one of the shortcuts in my textbook without any proof.<br>
When $b\gt a$, </p>
<blockquote>
<p>$$\int\limits_a^b \dfrac{dx}{\sqrt{(x-a)(b-x)}}=\pi$$</p>
</blockquote>
<hr>
<p><strong>My attempt :</strong></p>
<p>I notice that the the denominator is $0$ at both the bounds. I thought of su... | Nosrati | 108,128 | <p>Other way is substitution $t=\sin^2\theta$ so
$$\int\limits_0^1 \dfrac{dt}{\sqrt{t(1-t)}}=\int\limits_0^\frac{\pi}{2} 2dt=\pi$$</p>
|
2,853,673 | <p>I came across this as one of the shortcuts in my textbook without any proof.<br>
When $b\gt a$, </p>
<blockquote>
<p>$$\int\limits_a^b \dfrac{dx}{\sqrt{(x-a)(b-x)}}=\pi$$</p>
</blockquote>
<hr>
<p><strong>My attempt :</strong></p>
<p>I notice that the the denominator is $0$ at both the bounds. I thought of su... | Ng Chung Tak | 299,599 | <p>\begin{align}
\tan^2 \theta &= \frac{x-a}{b-x} \\
2\tan \theta \sec^2 \theta \, d\theta &=
\frac{b-a}{(b-x)^2} \, dx \\
2\sqrt{\frac{x-a}{b-x}} \times \frac{(x-a)+(b-x)}{b-x} \, d\theta &=
\frac{b-a}{(b-x)^2} \, dx \\
2\, d\theta &= \frac{dx}{\sqrt{(x-a)(b-x)}} \\
\int \frac{dx}{\sqrt{(... |
347,494 | <p>I have a question regarding differential forms.</p>
<p>Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?</p>
| Damien L | 59,825 | <p>The differential form $\omega = dx_1 \wedge dx_2$ is <em>constant</em> hence we have $$ d\omega = d(dx_1 \wedge dx_2) = d(1) \wedge dx_1 \wedge dx_2 \pm 1 \, ddx_1 \wedge dx_2 \pm 1 \, dx_1 \wedge ddx_2$$ and because $d^2 = 0$, we have $$ d \omega = 0.$$</p>
|
1,139,847 | <p>I am trying to solve a differential equation, but I can't solve an integral, because I forgot which rule to apply. What rule do I use to do $$\int \frac{1}{3y-y^2}\mathrm dy\ ?$$</p>
| Guadalupe | 214,011 | <p>You can use Partial Fraction Decomposition since $3y-y^2=y(3-y)$. So $\frac{1}{y(3-y)}=\frac{1/3}{y}+\frac{1/3}{(3-y)}$ and certainly you know the integral of $1/y$.</p>
<p>I hope I could help you :)</p>
|
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | Syed | 81,355 | <pre><code>res1 = data /. {{a_, b_, c_, d_} :> {b, c, d} /; a < 2 , {a_, b_, c_,
d_} :> Nothing}
</code></pre>
<p>Or phrased the other way round:</p>
<pre><code>res2 = data /. {{a_, b_, c_, d_} :>
Nothing /; a > 2 , {a_, b_, c_, d_} :> {b, c, d}}
</code></pre>
<p>Or</p>
<pre><code>res6 = Wh... |
4,565,003 | <p>I'm working on the following problem:</p>
<p>The holomorphic <span class="math-container">$1$</span>-form <span class="math-container">$\frac{dz}{1+z^2}$</span>,
defined on <span class="math-container">$\mathbb{C}\setminus\{\pm i\}$</span>, can be extended to a
holomorphic <span class="math-container">$1$</span>-for... | Quaere Verum | 484,350 | <p>I find your notation (using <span class="math-container">$\tan$</span> and <span class="math-container">$\sec$</span>) slightly confusing, so let me just work it out in the way that I consider to be "clean". We take the standard open cover <span class="math-container">$\{U_0,U_1\}$</span> for <span class="... |
4,565,003 | <p>I'm working on the following problem:</p>
<p>The holomorphic <span class="math-container">$1$</span>-form <span class="math-container">$\frac{dz}{1+z^2}$</span>,
defined on <span class="math-container">$\mathbb{C}\setminus\{\pm i\}$</span>, can be extended to a
holomorphic <span class="math-container">$1$</span>-for... | M.C. | 752,637 | <p>You also asked how one can find the pullback of the <span class="math-container">$1$</span>-form <span class="math-container">$\omega$</span> (on <span class="math-container">$\mathbb{P}^1\setminus\{\pm i\}$</span>) by the tangent function <span class="math-container">$\operatorname{tan}\colon \mathbb{C}\rightarrow ... |
755,989 | <p>I am looking for an intuitive explanation for the identity:</p>
<p>$$\binom{n}{h}\binom{n-h}{k} = \binom{n}{k}\binom{n-k}{h}$$</p>
<p>Thanks!</p>
| Euna | 140,573 | <p>$\binom{n}{h}$ means possible numbers of choosing $h$ objects out of $n$. So, possible numbers of choosing $h$ out of $n$ then choosing $k$ objects out of $n-h$ objects which are not chosen before would be $\binom{n}{h}\binom{n-h}{k}$. Similarly, choosing $k$ objects first and $h$ objects later out of $n$ would be $... |
755,989 | <p>I am looking for an intuitive explanation for the identity:</p>
<p>$$\binom{n}{h}\binom{n-h}{k} = \binom{n}{k}\binom{n-k}{h}$$</p>
<p>Thanks!</p>
| Graham Kemp | 135,106 | <p>The first counts the ways to choose $h$ objects from $n$ distinct objects and then $k$ objects from the remainder.</p>
<p>The second counts the ways to choose $k$ objects from $n$ distinct objects and then $h$ objects from that remainder. </p>
<p>The order of selection is irrelevant; both count the way to select $... |
1,803,843 | <p>Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property (epsilon-delta, sequential, oscillation, etc); a function is continuous on an interval if it is continuous at all $x \in I$. </p>
... | Masacroso | 173,262 | <p>Imagine any neighborhood of a point $x$. If every neighborhood intersect a set $A$ in a different point than $x$ then $x$ is a limit point of $A$. </p>
<p>To simplify: in the standard topology of $\Bbb R$ an $\epsilon$-neighborhood is any open interval of the kind $(x-\epsilon,x+\epsilon)$ for any $\epsilon>0$ (... |
3,603,281 | <blockquote>
<p>A cubic equation <span class="math-container">$x^{3}+ax^2+bx+c$</span> has all negative real roots and <span class="math-container">$a, b, c\in R$</span> with <span class="math-container">$a<3.$</span></p>
<p>Prove that <span class="math-container">$b+c<4.$</span></p>
</blockquote>
<p>My attempt :... | Davood | 477,916 | <p>We can write our polynomial in the form <span class="math-container">$(x+r_1)(x+r_2)(x+r_3)$</span>, where <span class="math-container">$r_i$</span> are positive real numbers. Note that <span class="math-container">$b=r_1r_2+r_2r_3+r_3r_1$</span> and <span class="math-container">$c=r_1r_2r_3$</span>.</p>
<hr>
<hr>... |
1,639,241 | <p>I'm using gradient descent with mean squared error as error function to do linear regression. Take a look at the equations first.
<a href="https://i.stack.imgur.com/GN90y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GN90y.png" alt="enter image description here"></a>
As you can see in eq.1, the... | Hagen von Eitzen | 39,174 | <p>$\{ceps_q\}_{q=0}^Q$ is the finite sequence (or <code>array</code> or <code>vector</code> in programmese) $$ceps_0,ceps_1,\ldots, ceps_Q.$$</p>
<p>Likewise, $\{a_q\}_{q=1}^p$ denotes $$a_1,a_2,\ldots, a_p.$$</p>
|
514,702 | <p>I know that this is true and is used to prove that $\mathbb{Q}$ is not a discrete metric space, but I can't figure out, why is it true ?</p>
| Nick Peterson | 81,839 | <p>A set $U$ in a metric space $(X,d)$ is open if, and only if, it contains a neighborhood around each of its points. That is, for any $a\in U$, you can find a radius $r>0$ so that $B(a,r):=\{y\in X\mid d(a,y)<r\}$ is contained in $U$.</p>
<p>So, for $\{a\}$ to be an open set in $\mathbb{Q}$ (under the usual me... |
33,369 | <p>I've been trying to get to grips with the various semantics commonly discussed in formal logic. Specifically, the nature and role of interpretations of first and higher-order logics is slightly unclear.</p>
<p>Here are a few of the specific questions that have occurred to me:</p>
<ul>
<li>Propositional logic only ... | Andrej Bauer | 1,176 | <p>Here are some further possibilites:</p>
<ul>
<li><p><em>Propositional logic:</em> can be interpreted in any Boolean algebra (assuming you are talking about classical logic here), but perhaps your notion of truth assignment allows for truth values as elements of Boolean algebras. Alternatively, via the <a href="http... |
2,008,656 | <p>$\lim_{x \to \infty }\sqrt[x]{a^{x}+b^{x}+c^{x}} $;</p>
<p>$a,b,c\in \mathbb{R}$.</p>
<p>I need to find and prove a limit of this sequence. </p>
<p>I know that for example the limit of $a^{x}$ is $\infty$ for $a>1$. And the limit of $\sqrt[x]{a^{x}+b^{x}+c^{x}}$ should be equal $\sqrt[x]{\lim_{x \to \infty }{... | Clement C. | 75,808 | <p>I assume $a,b,c > 0$.</p>
<p>For $x> 0$,
$$
\sqrt[x]{a^x+b^x+c^x}
=
\left(a^x+b^x+c^x\right)^{1/x}
= e^{\frac{1}{x} \ln(a^x+b^x+c^x) }.
$$
By continuity of the exponential, it is sufficient to compute the limit of the exponent, $\frac{1}{x} \ln(a^x+b^x+c^x)$, when $x\to \infty$.</p>
<p>Without loss of gener... |
2,824,411 | <p>My first thought was successful: <span class="math-container">$x^4+x^2=x^2(x^2+1)$</span> and <span class="math-container">$x^3+x^2+1=x^2(x+1)+1$</span> so it is its own inverse because <span class="math-container">$(x^2(x+1)+1)^2\equiv x^4(x+1)^2+1\equiv x^4(x^2+1)+1\equiv1.$</span></p>
<p>The given solution claims... | Phil H | 554,494 | <p>For any 4 cards from 10 there are $^{10}C_4 = 210$ combinations.</p>
<p>For any 4 cards from 52 there are $^{52}C_4 = 270725$ combinations.</p>
<p>$P(4A) = \frac{^{10}C_4}{^{52}C_4} = \frac{210}{270725} = .0007757$</p>
<p>In other words, your $10$ cards only comprises of $210$ four card combinations from a possib... |
2,824,411 | <p>My first thought was successful: <span class="math-container">$x^4+x^2=x^2(x^2+1)$</span> and <span class="math-container">$x^3+x^2+1=x^2(x+1)+1$</span> so it is its own inverse because <span class="math-container">$(x^2(x+1)+1)^2\equiv x^4(x+1)^2+1\equiv x^4(x^2+1)+1\equiv1.$</span></p>
<p>The given solution claims... | Graham Kemp | 135,106 | <p>Yes, the probability for selecting four specific cards (and six others) when selecting ten cards from a standard 52 card deck (without replacement) is $${\left.{\dbinom 44}\dbinom{48}6\middle/\dbinom{52}{10}\right.}$$</p>
<p>The probability that those four cards shall be placed among the top ten places in the deck ... |
2,539,942 | <p>For any $x\in\mathbb{R}$, the series
$$ \sum_{n\geq 1}\tfrac{1}{n}\,\sin\left(\tfrac{x}{n}\right) $$
is trivially absolutely convergent. It defines a function $f(x)$ and <strong>I would like to show that $f(x)$ is unbounded over $\mathbb{R}$</strong>. Here there are my thoughts/attempts:</p>
<ol>
<li>$$(\mathcal{L}... | Jack D'Aurizio | 44,121 | <p>Thanks to MartinR and HJol for referencing <a href="http://onlinelibrary.wiley.com/doi/10.1112/plms/s2-41.4.257/full" rel="noreferrer">a classical result of Hardy and Littlewood</a>, which I am going to outline. The final result is </p>
<blockquote>
<p>$$ \sum_{n\geq 1}\frac{1}{n}\,\sin\frac{x}{n}=\Omega\left(\sq... |
2,023,400 | <p><span class="math-container">$\textbf{Question:}$</span> Find a basis for the vector space of all <span class="math-container">$2\times 2$</span> matrices that commute with <span class="math-container">$\begin{bmatrix}3&2\\4&1\end{bmatrix}$</span>, which is the matrix <span class="math-container">$B$</span>.... | Marc van Leeuwen | 18,880 | <p>Adapted from <a href="https://math.stackexchange.com/a/1963802/18880">this answer</a> to a very similar question.</p>
<p>That matrix $B$ is clearly not a multiple of the identity matrix, so its minimal polynomial is of degree${}>1$, hence equal to its characteristic polynomial (which you do not have to compute).... |
4,253,330 | <p>Show that the sequence of functions <span class="math-container">$\langle f_n \rangle$</span> defined by <span class="math-container">$$f_n(x) = \frac{nx}{nx+1} \, ; \, n \in \mathbb{N}$$</span> fails to converge uniformly on <span class="math-container">$[0, \infty) .$</span></p>
<hr />
<p>The name of the game here... | José Carlos Santos | 446,262 | <p>The sequence <span class="math-container">$(f_n)_{n\in\Bbb N}$</span> converges pointwise to <span class="math-container">$1$</span>. Therefore, if it converged uniformly on <span class="math-container">$[0,\infty)$</span>, it would converge uniformly to <span class="math-container">$1$</span>. But<span class="math-... |
3,223,371 | <p>In order to prove if a relation is an equivalence relation, it needs to be show that is all of: </p>
<ul>
<li>Reflexive </li>
<li>Symmetric </li>
<li>Transitive </li>
</ul>
<p>Whilst I am familiar with this, I am unsure how to approach the following set of questions: </p>
<p>State and explain whether each of... | Ethan Bolker | 72,858 | <p>You can get around the "cartesian products of cartesian products" confusion by writing the relation the way you write "<span class="math-container">$=$</span>", so
<span class="math-container">$$
(a,b) R (c,d) \iff ad = bc .
$$</span>
Now the only cartesian product involves the pairs that may or may not be related. ... |
2,912,881 | <p>In an $n\times n$ board ($n\geq 3$), how many colors do we need so that we can color the cells such that no three consecutive cells (horizontal, vertical, or diagonal) are of the same color?</p>
<p>With three colors we can do it, using the pattern</p>
<p>$$131$$
$$232$$
$$312$$</p>
<p>and repeating it as necessar... | ArsenBerk | 505,611 | <p><strong>HINT:</strong> In total, there are $4$ upward moves ($U$) and $4$ rightward moves ($R$). So for example $UUUURRRR$ is one of the ways ant can use. So the problem can be restated as "How many different words (meaningful or not meaningful) can be formed with $4\ U$'s and $4\ R$'s?".</p>
|
2,735,007 | <p>The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative method published in 1970 by Michael A. Jenkins and Joseph F. Traub. </p>
<p>The published algorithm, e.g. <a href="https://dl.acm.org/citation.cfm?id=355643" rel="nofollow noreferrer">https://dl.acm.org/citation.cfm?id=355643</... | user7530 | 7,530 | <p>No chance. Solving for the roots of multivariate polynomials is a completely different beast from the univariate case.</p>
<p>For one thing, a multivariate polynomial will typically be zero on entire curves, and not just at isolated points (consider dxiv's $x^2+y^2-1$ for instance). So usually one seeks to reduce <... |
4,626,092 | <p>Question:
If the wire of length is cut into three pieces then the probability that the three pieces form a triangle is ____</p>
<p>My approach is as follow</p>
<p>For a triangle to exist the sum of two sides should be greater than the third side</p>
<p>Hence <span class="math-container">$x + y + z = \ell \Rightar... | gnasher729 | 137,175 | <p>You’d have to describe the process of “cutting in pieces” precisely.</p>
<p>I cut the wire at a random point with linear distribution. If I then cut the smaller piece, it’s impossible to get a triangle. If I cut the longer piece then I get some probability, if I first mark the complete wire in two places and cut at ... |
4,442,117 | <p>This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution with parameter β. Determine the expected length of time until three items have failed"</p>
<p>I approached the pr... | lpz | 1,051,191 | <p>Your reasoning only works in <span class="math-container">$2-$</span>D, and this is why it works only for <span class="math-container">$b=0$</span>. However, since you calculated <span class="math-container">$T$</span>, you'll notice it traces out a circle, and the vector sweeps out a cone. Actually, the distance tr... |
3,559,942 | <p>I am trying to solve the limit:
<span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)$$</span></p>
<p>I was trying to find a way to bring it into a fraction form to apply L'Hospital's rule, and I tried using
<span class="math-... | K.K.McDonald | 302,349 | <p><span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)= \lim_{x\to\infty}x^2\left(\left(1+\frac{1}{x}\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-1\right)=\lim_{x\to\infty}\frac{\left(1+\frac{1}{x}\sin\left(\frac{1}{x}\right... |
354,961 | <p>Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Show that $A\cap B$ is a normal subgroup of $B$.</p>
| Martin Brandenburg | 1,650 | <p>$B \cap A$ is the kernel of $B \to G \to G/A$.</p>
|
1,853,306 | <blockquote>
<p>Consider the relations R and S on <span class="math-container">$\Bbb N$</span> defined by <span class="math-container">$x\; R\; y$</span> iff</p>
<p><span class="math-container">$2 \;$</span>divides <span class="math-container">$x + y$</span>
and <span class="math-container">$x \;S \;y$</span> iff <sp... | parsiad | 64,601 | <p>The product of any two orthonormal matrices is also orthonormal. Let
$L,X$ be two such matrices. Then, $(LX)^{*}(LX)=X^{*}L^{*}LX=I$.</p>
<p>To answer your question, yes: pick $X$ to be any (real) orthonormal matrix not equal to $L^*$ or the identity.</p>
|
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | Mythomorphic | 152,277 | <p>It would be better to say</p>
<p>As $0\le|{\cos\theta}|\le1$,</p>
<p>$0^2\le|{\cos\theta}|^2\le1^2$</p>
<p>$0\le{\cos^2\theta}\le1$</p>
|
53,798 | <p>Let $f(i),i\in \mathbb N\, $ be a sequence of real or complex numbers then for natural numbers $m,n$ and $r$ holds sum transformation</p>
<p>$$\sum_{i=0}^{mn+r}f(i)=\sum_{i=0}^{r}f(mn+i)+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}f(mj+i).$$</p>
<p>This identity can be proved by induction by $r$. I am looking for an alternati... | Chris Taylor | 4,873 | <p>You need to check that the set</p>
<p>$$\{0,1,\dots,mn+r\}$$</p>
<p>is identical to the set</p>
<p>$$\{mn+i : 0\leq i\leq r\} \cup \{mj+i : 0\leq i < m, 0\leq j < n\}$$</p>
<p>This is not too hard: the second set in the union counts all of the numbers from $0$ to $mn-1$, and the first set counts all of the... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | MaxW | 23,782 | <p>As has been pointed out $x^2-5x+6=0\quad$ is not the equation of the parabola. </p>
<p>The equation of the parabola is:</p>
<blockquote>
<p>$y_p = x^2-5x+6 = (x-2)(x-3)$ </p>
</blockquote>
<p>The equations of the two straight lines within the equation are is $y_1 = x-2$ and $ y_2 = x - 3$ and both lines can be ... |
2,960,132 | <p>Prove or disprove each of the follow function has limits <span class="math-container">$x \to a$</span> by the definition </p>
<p><span class="math-container">$\lim_{(x, y) \to (0, 0)} \frac{x^2y}{x^2 + y^2}$</span> </p>
<p>Let <span class="math-container">$y = x^2$</span></p>
<p><span class="math-container">$... | Theo Bendit | 248,286 | <p>Recall the Mazur-Ulam theorem: a surjective isometry between normed linear spaces must be an affine map. There is a Wikipedia stub on the subject, but it's not very helpful. Check out <a href="http://www.helsinki.fi/~jvaisala/mazurulam.pdf" rel="nofollow noreferrer">this short and rather elegant proof</a> instead.</... |
683,970 | <p>Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map.</p>
<p>I'm trying to prove that there exists only one $z_0 \in H$ such that:<br>
$ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$</p>
<p>Proof: </p>
<p>If $f \equiv 0 $ then we have $z_0 = 0$. </p>
<p>If $f \not\equiv 0$ then:<br>
Let $... | J.R. | 44,389 | <p>$f$ is continuous, because its bounded, so $K=f^{-1}(\{0\})$ is closed as preimage of a closed set under a continuous map.</p>
|
2,537,344 | <p>I'm trying to find an expression for $\prod_{j=1}^n\left(j-\frac{1}{\sqrt{j}}+1\right)$ in terms of n. I have tried finding $\int_1^n\ln\left(x-\frac{1}{\sqrt{x}}+1\right)\,dx$ by parts but end up with a big mess. Any help would be appreciated!</p>
| robjohn | 13,854 | <p>Since
$$
\log\left(x-\frac1{\sqrt{x}}+1\right)=\log(x+1)+\log\left(1-\frac1{(x+1)\sqrt{x}}\right)\tag1
$$
we can expand
$$
\begin{align}
\log\left(1-\frac1{(x+1)\sqrt{x}}\right)
&=-x^{-3/2}+x^{-5/2}-\frac12x^{-3}-x^{-7/2}+x^{-4}+\frac23x^{-9/2}-\frac32x^{-5}\\
&+\frac74x^{-6}-x^{-13/2}-\frac32x^{-7}+\frac{32... |
330,236 | <p>I'm dutch and I'm not sure if I translated this right. If there are some more dutchies here, how could I translate: "volledig stelsel van representanten"</p>
<p>Let $H$ be a subgroup of of $G$ and consider the left cosets of $H$. Pick from every distinct left coset one element, and put them in the set $R$. Show tha... | yatima2975 | 360 | <p>First off, I have to express my concern about whatever happened to $G_2$!</p>
<p>The proof provided looks completely right to me. If you want to be completely explicit, you could include some additional steps, e.g. after the first sentence you could insert </p>
<p>"So $a\in G_1$ and $b \in G_1$, therefore $x = ab ... |
4,479,797 | <p>Let's say I have an urn with 10 unique objects, and I choose 3 objects from it (each choice is made without replacement). Then the probability of choosing any one object is 3/10. I calculated this probability by summing the probability the object is chosen on the first pick + probability chosen on second pick + prob... | Claude Leibovici | 82,404 | <p>You can do it faster since
<span class="math-container">$$\frac {y^2}{1+y^4}=\frac {y^2}{(y^2+i)(y^2-i)}=\frac 1 2\Big[\frac 1{y^2+i} +\frac 1{y^2-i}\Big]$$</span> and
<span class="math-container">$$\int \frac {dy}{y^2+k\, i}=-\frac{1-i}{\sqrt{2k}}\tan ^{-1}\left(-\frac{1-i}{\sqrt{2k}}{y}\right)$$</span></p>
|
1,397,036 | <p>A bag labeled $A$ contains $4$ red balls and $7$ green balls.
Another bag $B$ contains $6$ red and $5$ green balls.</p>
<p>A ball is transferred from bag $A$ to bag $B$, after which a ball is drawn from $B$.</p>
<p>Find the probability that it is a red ball?</p>
<p>To be honest I have no idea how to approach the ... | GFauxPas | 173,170 | <p>Hint: there are two cases to consider. One case is that a green ball is moved to bag $B$. The other case is a red ball is moved to bag $B$.</p>
|
1,397,036 | <p>A bag labeled $A$ contains $4$ red balls and $7$ green balls.
Another bag $B$ contains $6$ red and $5$ green balls.</p>
<p>A ball is transferred from bag $A$ to bag $B$, after which a ball is drawn from $B$.</p>
<p>Find the probability that it is a red ball?</p>
<p>To be honest I have no idea how to approach the ... | Amaury | 262,046 | <p><strong>Hint:</strong></p>
<p>Think step by step to come back to cases you know.</p>
<ol>
<li>First step: Transfer a ball</li>
</ol>
<p>How many different cases are they ? What is the probability of each case ?</p>
<ol start="2">
<li>Second step: for each case of the previous step: draw a ball</li>
</ol>
<p>Wha... |
547,932 | <p>I have </p>
<p>$f(x)=\sqrt{3x}+1$</p>
<p>$g(x)=x+1$</p>
<p>My thinking was that at the intersection points both will be equal to each other so </p>
<p>$\sqrt{3x}+1=x+1$</p>
<p>$\sqrt{3x}=x$</p>
<p>However I don't know where to go from here.</p>
| amWhy | 9,003 | <p>$$\sqrt{3x} = x \implies 3x = x^2 \iff x^2 - 3x = x(x - 3) = 0$$
$$\implies x = 0 \quad \text{OR}\quad x = 3$$</p>
<p>Hence, the intersection points will be $(0, f(0)),$ and $(3, f(3))$.</p>
|
4,214,550 | <p>Before stating the claim, let my define all the objects which I need: To start with, let me fix notation:</p>
<ul>
<li>Let <span class="math-container">$P$</span> be a principal <span class="math-container">$G$</span>-bundle over a (smooth, compact, oriented) manifold <span class="math-container">$\mathcal{M}$</span... | Yuri Vyatkin | 2,002 | <p>Let me restate everything in the <a href="https://math.stackexchange.com/a/193587/2002">abstract index notation</a>, preserving some of the notation from the question, but adapting where it makes more sense to me. In particular, I try to keep the indices balanced.</p>
<p>The definition of the wedge product can be re... |
4,214,550 | <p>Before stating the claim, let my define all the objects which I need: To start with, let me fix notation:</p>
<ul>
<li>Let <span class="math-container">$P$</span> be a principal <span class="math-container">$G$</span>-bundle over a (smooth, compact, oriented) manifold <span class="math-container">$\mathcal{M}$</span... | G. Blaickner | 955,008 | <p>For completeness, let me post an answer to my question. First of all, the claim I wanted to proof is correct. As pointed out in the comment by @levap, I forgot to take the derivative of <span class="math-container">$\langle e_{a},e_{b}\rangle_{E}\in C^{\infty}(U)$</span> on the left-hand side. It follows that the cl... |
3,305,140 | <p>So, I'm going to take an enumerative combinatorics class this upcoming semester. I began reading about it and came across and interesting example, but I am not sure how they arrive at their final answer. The example is in in the image I included. I don't know how they determined the equations for <span class="math... | Christian Blatter | 1,303 | <p>The maximal <span class="math-container">$(t,u)$</span>-domain relevant to the given IVP is <span class="math-container">$\Omega:={\mathbb R}_{>0}\times{\mathbb R}$</span>. Within <span class="math-container">$\Omega$</span> the standard existence and uniqueness theorem for ODEs is valid. By inspection one sees t... |
1,006,445 | <p>Prove $(0,1)$ and $[0,1]$ have the same cardinality. </p>
<p>I've seen questions similar to this but I'm still having trouble. I know that for $2$ sets to have the same cardinality there must exist a bijection function from one set to the other. I think I can create a bijection function from $(0,1)$ to $[0,1]$, ... | gamma | 88,524 | <p>Use Hilbert's Hotel. </p>
<p>First identify a countable subset of $(0,1)$, say $H = \{ \frac1n : n \in \mathbb N\}$.</p>
<p>Then define $f:(0,1) \to [0,1]$ so that </p>
<p>$$ \frac12 \mapsto 0$$
$$ \frac13 \mapsto 1$$
$$ \frac{1}{n} \mapsto \frac{1}{n-2}, n \gt 3$$
$$ f(x) = x, \text{for } x \notin H $$</p>
|
4,233,449 | <blockquote>
<p>Consider <span class="math-container">$\mathcal F = \{f$</span> holomorphic in <span class="math-container">$\Bbb D$</span> with <span class="math-container">$f(\Bbb D)\subset\Bbb D,$</span> <span class="math-container">$f\left(\frac12\right)=f'\left(\frac12\right)=0\}$</span>, where <span class="math-c... | Gerd | 960,044 | <p>Consider the Blaschke factor <span class="math-container">$T(z)=(z+1/2)/(1+z/2)$</span>. Let <span class="math-container">$f \in {\cal F}$</span>
and set <span class="math-container">$g(z):=f(T(z))$</span>. Then <span class="math-container">$g(\mathbb{D}) \subseteq \mathbb{D}$</span>, <span class="math-container">$g... |
301,198 | <p>I am new to linear algebra and I have a doubt that : in 2D coordinate system is a line which is at 45 degree <strong>NOT</strong> passing through the origin a subspace of the vector space comprising the whole 2D plane i.e. $ \mathbb{R}^2 $ ?
let $V = \{ (x,y) \in \mathbb{R}^2 \}$. and $W = \{ (x,y) \in \mathbb{R}^... | GEdgar | 442 | <p>Your $W$ is not a <strong>linear subspace</strong>, which is what we mean when we say "subspace" in a linear algebra course. But $W$ is an <strong>affine subspace</strong>, a term used in mathematics as well.</p>
|
1,378,960 | <blockquote>
<p>If $3\sin A + 5\cos A = 5$, then prove that:
$$5\sin A + 3\cos A = ±3.$$</p>
</blockquote>
| David Quinn | 187,299 | <p>HINT. Try squaring the first identity and then use $\sin^2A+\cos^2A=1$</p>
|
1,725,343 | <p>I found the quadratic approximation as $9 + \frac{1}{2}(-9x^2 - 9y^2)$</p>
<p>The problem is that the triple derivatives all end up 0 at (0,0), so I get that the error approximation is 0. Wolfram alpha calculates the triple derivatives having sin(y) or sin(x) in them making them 0. I know you are supposed to plug i... | André Nicolas | 6,312 | <p>Hint: To find a series expression for $\frac{2}{(8+x)^2}$, differentiate the power series of (more or less) $\frac{1}{8+x}$. Note that $\frac{1}{8+x}$ has derivative $-\frac{1}{(8+x)^2}$.</p>
<p>To find the series for $\frac{1}{8+x}$, rewrite as $\frac{1}{8}\cdot \frac{1}{1+x/8}$, and use the familiar series for $\... |
1,450,438 | <p>So the full problem was: </p>
<blockquote>
<p>Consider $R=\mathbb Q[x,y,z]/(x-xyz)$. Prove that $x$ and $xy$ divide each other in $R$ but that they are not associates. In other words, there is no unit $u\in R$ so that $x=xyu$. </p>
</blockquote>
<p>I already proved they divide each other but I need to prove no u... | Lubin | 17,760 | <p><strong>EDIT</strong>: After a helpful observation and correction from user26857, I have made significant changes in the fourth paragraph below (now the fifth!), related to the units of $A$ and those of $B$. I’m afraid that my proof is now a bit messier.</p>
<p>As I suggested in my comments, the key to this questio... |
1,450,438 | <p>So the full problem was: </p>
<blockquote>
<p>Consider $R=\mathbb Q[x,y,z]/(x-xyz)$. Prove that $x$ and $xy$ divide each other in $R$ but that they are not associates. In other words, there is no unit $u\in R$ so that $x=xyu$. </p>
</blockquote>
<p>I already proved they divide each other but I need to prove no u... | user26857 | 121,097 | <p>As the answer in the linked thread shows us, <em>there is no need to find the units of $R$ in order to solve the problem</em>. But if we want to do this, then the nicest way I know is the following (instead of $\mathbb Q$ let's consider an arbitrary field $k$):</p>
<p>There is an injection
$$R \hookrightarrow k[X,Y... |
4,096,885 | <p>For example if I have two sets <em>A</em> and <em>B</em>, where I take the Cartesian Product of both, does it matter if I perform the operation in this order <em>A</em>x<em>B</em> or whether I perform the operation in this order or the order <em>B</em>x<em>A</em>, or does the order not even matter? I am enquiring be... | Community | -1 | <p>The order certainly matters, namely <span class="math-container">$A=\{x\,:\, \exists y, (x,y)\in A\times B\}$</span> and <span class="math-container">$B=\{x\,:\, \exists y, (y,x)\in A\times B\}$</span>. While there are instances where this could be treated as a technicality (and the longer I think about them, the fe... |
2,243,598 | <p>Consider this integral $(1)$</p>
<blockquote>
<p>$$\int_{0}^{\infty}\color{red}{{\gamma+\ln x\over e^x}}\cdot{1-\cos x\over x}\,\mathrm dx={1\over 2}\cdot{\pi-\ln 4\over 4}\cdot{\pi+\ln 4\over 4}\tag1$$</p>
</blockquote>
<p>Recall a well-known integral for $\gamma$:</p>
<p>$$\int_{0}^{\infty}e^{-x}\ln x\,\mathr... | Random Variable | 16,033 | <p>Assuming that $a>1$,</p>
<p>$$ \begin{align} I(a) &= \int_{0}^{\infty} \frac{\gamma+\ln x}{e^{ax}} \frac{1-\cos x}{x} \, dx \\ &= -\int_{0}^{\infty} \frac{\gamma+\ln x}{e^{ax}} \sum_{n=1}^{\infty} \frac{(-1)^{n}x^{2n-1}}{(2n)!} \, dx \\ &= -\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2n)!} \int_{0}^{\infty... |
550,817 | <p><img src="https://i.stack.imgur.com/d4kwN.png" alt="enter image description here"></p>
<p>I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though.</p>
<p>I was looking at this question for a hint, but I... | xavierm02 | 10,385 | <p>Your problem here is probably that you think of matrices as matrices(or linear transformations) and not as vectors.</p>
<p>You decompose your matrix $M=\begin{pmatrix}a&b\\0&c\end{pmatrix}$ and decompose it as $M=a\begin{pmatrix}1&0\\0&0\end{pmatrix}+b\begin{pmatrix}0&1\\0&0\end{pmatrix}+c\b... |
74,086 | <p>Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$.</p>
<p>Then for any $l\geq 1$, the projective scheme $\operatorname{Quot}(E,l)$ classifies quotients $E\to T$, such that $\operatorname{dim}_{\mathbb{C}}(H^0... | Igor Rivin | 11,142 | <p>Yes. See this paper:</p>
<p>www.intlpress.com/JOC/p/2010/JOC-1-2-a1-Thomassen.pdf</p>
<p>(Carsten Thomassen, spanning trees and orientations in graphs; google shows you the full text...), and references therein (esp. McKay's paper [9, op cit])</p>
<p>He treats cubic graphs, which is about as sparse as you can get... |
1,077,248 | <p>Prove:
For every positive integer $n, n^2 + 4n + 3$ is not a prime.</p>
<p>I tried to disprove the statement, which I could not using several number examples with constructive proof.</p>
<p>However I am not sure how to correctly step by step prove it.</p>
<p>Thank you in advance!</p>
| DeepSea | 101,504 | <p>Hint: $n^2+4n+3 = (n+1)(n+3)$. Can you take it further?</p>
|
1,077,248 | <p>Prove:
For every positive integer $n, n^2 + 4n + 3$ is not a prime.</p>
<p>I tried to disprove the statement, which I could not using several number examples with constructive proof.</p>
<p>However I am not sure how to correctly step by step prove it.</p>
<p>Thank you in advance!</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is $$(n+1)(n+3)$$ and thus not prime because it has to factors greater than one.</p>
|
3,289,999 | <p>How can i easily determine the order of
<span class="math-container">$$
[3] \in \mathbb{F}_{11}^\times
$$</span></p>
<p>By the way: <span class="math-container">$\mathbb{F}_{11}^\times =\mathbb{F}_{11}\setminus \{[0]\}$</span>.</p>
<p>Fermat's little theorem states that the order of a group element
has to be a di... | J. W. Tanner | 615,567 | <p>Your answer is correct.</p>
<p><strong>Hint for a faster way:</strong> <span class="math-container">$5^2\equiv3\mod11$</span>, and by Fermat's little theorem <span class="math-container">$5^{10}\equiv1\mod11.$</span></p>
|
4,093,001 | <p>Show that if <span class="math-container">$X$</span> is path connected then every path <span class="math-container">$f:I\to X$</span> is homotopic to a constant path <span class="math-container">$g(t)=x$</span>. Does this mean show that there is an <span class="math-container">$x$</span> (or even for all <span class... | Henno Brandsma | 4,280 | <p>This can work (to assuage your doubts) as the homotopy is a "free" one, where the homotopy does not have to preserve any fixed point, as in the fundamental group, which is why the torus or circle don't work as counterexamples..</p>
<p>If <span class="math-container">$f$</span> is a path, we can define <spa... |
4,351,777 | <p>I'm studying about tensors and have already understood the following theorem:</p>
<blockquote>
<p><span class="math-container">$C^1_1$</span> is the unique linear function such that <span class="math-container">$C_1^1(v\otimes\eta)=\eta(v)=v(\eta)$</span> for all <span class="math-container">$v\in V$</span> and <spa... | Kurt G. | 949,989 | <p>Any linear map <span class="math-container">$C$</span> between two vector spaces <span class="math-container">${\cal U}$</span> and <span class="math-container">${\cal W}$</span> is uniquely defined as long as it is defined on a basis of <span class="math-container">${\cal U}\,.$</span> A basis of <span class="math-... |
159,775 | <p>Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure?</p>
<p>By a Physical measure i mean in the sense of $\nu$ a hyperbolic(non-atomic) measure then having positive Lebesgue Basin of attraction.
&
By an SRB measure i mean in the sense ... | Rafael Alcaraz Barrera | 10,518 | <p>Maybe the best reference to clarify that is L.S. Young's paper: <a href="http://link.springer.com/article/10.1023/A:1019762724717" rel="noreferrer">What are SRB Measures, and Which Dynamical Systems have them.</a> Several autors mention them to be the same. In the comments, Pengfei give you a good example when they ... |
1,099,885 | <p>How can you calculate</p>
<p>$$\lim_{x\rightarrow \infty}\left(1+\sin\frac{1}{x}\right)^x?$$</p>
<p>In general, what would be the strategy to solving a limit problem with a power?</p>
| Community | -1 | <p>Use Taylor expansion $\sin t=t (1+o(1))$ when $t\to 0$. Then
$\left(1+sin \frac{1}{x}\right)^x \sim \left(1+\frac{1}{x}\right)^x\to e$.</p>
|
4,473,931 | <p>Swedish Lotto rules can be found <a href="https://cdn1.svenskaspel.net/content/cms/documents/779afe3f-0363-4c36-b79c-1881549a8cbc/1.13/spelregler-lotto-och-joker.pdf" rel="nofollow noreferrer">here</a>. As far as I know these are the most recent rules. In several places the probability of winning Drömvinsten is quot... | mikaru9221 | 1,068,285 | <p><strong>JMoravitz</strong> answerd my question in his two comments above:</p>
<p>Your attempted calculations seem to ignore the possibility of the ending digits outperforming the leading digits as well as your attempted calculations seem to ignore the possibility of the ending digits performing just as well as the l... |
1,124,078 | <p>Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.</p>
| shunjie | 106,203 | <p>Or just take $x=0$, then $\log 2 \ne \log 4$. </p>
|
194,642 | <p>Given a poset $(P,\leq)$ the <em>interval topology</em> on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.</p>
<p>Let $\{P_i : i\in I\}$ be a family of posets such that the interval... | Dominic van der Zypen | 8,628 | <p>The answer is <strong>Yes</strong>, because the product topology of Hausdorff spaces is Hausdorff again, and because the product topology of the interval topologies of a family of posets equals the interval topology of the product:</p>
<p><strong>Lemma.</strong> The interval topology $\tau_i = \tau_{\textrm{int}}(\... |
2,154,513 | <p>Suppose that $U$ is an open set containing $0.$ $f,g:U\to\mathbb{R}$ are continuos functions such that</p>
<ul>
<li>$g\in C^1(U)$</li>
<li>$g(0)=0$</li>
<li>$f|_{U\setminus 0}\in C^1(U\setminus 0)$ ($f$ is of $C^1$ class away of $0$)</li>
</ul>
<blockquote>
<p><strong>Question</strong>. Does product $fg$ is of $... | Andrew D. Hwang | 86,418 | <p><strong>Hint</strong>: The function
$$
h(x) = \begin{cases}
x^{2}\sin(1/x) & x \neq 0, \\
0 & x = 0,
\end{cases}
$$
is well-known (and easily seen) to be differentiable, but not $C^{1}$ at $0$. Factorizations are easily found that give counterexamples of the type you seek.</p>
|
95,176 | <p>Does the octic,</p>
<p>$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$</p>
<p>for any constant <em>n</em> have Galois group of order 1344? Its discriminant <em>D</em> is a perfect square,</p>
<p>$D = (1728n^4-341901n^3-11560361n^2+3383044089n+28121497213)^2$</p>
<p>Surely (1) is not an isolated result. ... | Alastair Litterick | 3,935 | <p>More of a comment than an answer, but there are exceptions for $n = 17$ and $n = 145$, where the Galois group is simple of order 168. This is all the exceptions for $|n| < 2\times10^5$.</p>
|
30,728 | <p>Is this graph in the list among the so-called "standard" structures used in <code>GraphData</code>? However, I have not found yet anything like "Carpet" or "Sponge" in the list of the objects that can be built. Maybe, this graph has a different name? </p>
<p>For me, using <code>GraphData</code> helps to save time f... | Dr. belisarius | 193 | <p>Here is a "general" <a href="http://en.wikipedia.org/wiki/L-system" rel="nofollow noreferrer">Lindenmayer System</a> generator I wrote <a href="https://codegolf.stackexchange.com/a/9351/315">in the spirit of code-golf</a>. Please beware that the objective of code-golfing is writing the shorter possible pr... |
3,288,475 | <p>Is there any formula to calculate anti logarithm just using simple calculator.. I already know how to calculate logarithm digit by digit exactly just like this <a href="https://math.stackexchange.com/questions/820094/what-is-the-best-way-to-calculate-log-without-a-calculator">What is the best way to calculate log wi... | Ross Millikan | 1,827 | <p>What you have to prove is that if I give you a <span class="math-container">$b$</span> that is a multiple of <span class="math-container">$4$</span>, you can express it as <span class="math-container">$\frac {s^2-t^2}2$</span> with <span class="math-container">$s,t$</span> odd and coprime. As you did, write <span c... |
344,479 | <p>Suppose that $a = 2^kb,$ where $b$ is odd. If $\phi(x) = a,$ prove that $x$ has at most $k$ odd prime divisors.</p>
| Stefan Babos | 66,538 | <p>I believe that I can prove that such $k \gt 0$ does not exist.<br/>
If $a$ has only odd divisors, then we can express $a$ as product these odd numbers. Product odd numbers is always odd number($(2u+1)(2v+1)=4uv+2u+2v+1=2z+1$-is odd). Number $2^kb$ can be odd only if $k=0$ and $b$ is odd, for $k\gt0$ is $2^kb$ always... |
4,172,689 | <p>I have this limit as my question to solve:</p>
<p><span class="math-container">$$\lim_{x\to-1}\frac{2x+\sqrt{3-x}}{x^2+x}$$</span></p>
<p>My procedure:</p>
<p><span class="math-container">$$\lim_{x\to-1}\frac{(2x+\sqrt{3-x})(2x-\sqrt{3-x})}{(x^2+x)(2x-\sqrt{3-x})}$$</span></p>
<p><span class="math-container">$$\lim_... | José Carlos Santos | 446,262 | <p>It's because <span class="math-container">$\displaystyle4x^2+x-3=(x+1)(4x-3)$</span> rather than <span class="math-container">$\displaystyle(x+1)\left(x-\frac34\right)$</span>.</p>
|
1,578,717 | <p>What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such expansion legal? </p>
| Ron Gordon | 53,268 | <p>To answer your question: it is not a Maclurin expansion of this function, or the integrand of the ILT. Such an expansion does not exist because of the essential singularity at $s=0$. Rather, as I will demonstrate below, it is a Laurent expansion of the integrand. A Laurent expansion of a function about an essenti... |
194,813 | <p>Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$</p>
| Thomas Andrews | 7,933 | <p>Let $f(u)=a^u$. Then the expression inside the limit is $\frac{f(z)-f(y)}{z-y}$ where $z=a^x$ and $y=x^a$.</p>
<p>So, by the mean value theorem, this expression is $f'(c)$ for some $c$ between $z$ and $y$.</p>
<p>But as $x\to a$, both $z$ and $y$ approach $a^a$, and therefore $c$ approaches $a^a$. Since $f'(u)$ i... |
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