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 |
|---|---|---|---|---|
1,049,808 | <p>Let $u_0 = 1$ and $u_{n+1} = \frac{u_n}{1+u_n^2}$ for all $n \in \mathbb{N}$.</p>
<p>I can show that $u_n \sim \frac{1}{\sqrt{2n}}$, but I would like one more term in the asymptotic development, something like $u_n = \frac{1}{\sqrt{2n}}+\frac{\alpha}{n\sqrt{n}} + o\bigl(\frac{1}{n^{3/2}}\bigr)$.</p>
<p>Here is the... | G.Kós | 141,614 | <p>Let $a_n=\dfrac1{u_n^2}$. Then
$$
a_{n+1}-a_n = \left(\frac{1+u_n^2}{u_n}\right)^2 - \frac1{u_n^2} = 2+u_n^2 = 2+\frac1{a_n}.
$$
From this and $a_2=4$ we can find a successive sequence of estimates for $n\ge2$:
\begin{align*}
a_n &\ge 2n; \\
a_n &\le 2n + \sum_{k=2}^{n-1}\frac1{2k}
< 2n+\frac12\lo... |
674,620 | <p>What is the slope of the tangent line to the function $$g(x)=x^2 \frac{\cos x}{1+x^3}$$ when $x=\pi/2$?
When $x=a$?</p>
| Semsem | 117,040 | <p>First find the derivative
$$g'(x)= \{\frac{x^2\cos x}{1+x^3}\}'
\\=\frac{(1+x^3)(2x\cos x-x^2\sin x)-(x^2\cos x)(3x^2)}{(1+x^3)^2}$$
Second the slope is $m=g'(\frac{\pi}{2})$</p>
|
1,549,843 | <p>Prove the following limits:</p>
<p>$$\lim_{x \rightarrow 0^+} x^x = 1$$
$$\lim_{x \rightarrow 0^+} x^{\frac{1}{x}}=0$$
$$\lim_{x \rightarrow \infty} x^{\frac{1}{x}}=1$$</p>
<p>They are not that hard using l'Hospital or the Sandwich theorem. But I curious if they can be solved with the basic knowledge of limits. I ... | Hagen von Eitzen | 39,174 | <p>Using $a^b=\exp(b\ln a)$ as definition of exponentiation with irrational exponents, it is natural to take logarithms; then the claims are equivalent to
$$\lim_{x\to 0^+}x\ln x=0,\qquad \lim_{x\to 0^+}\frac 1x\ln x=-\infty, \qquad \lim_{x\to \infty}\frac 1x\ln x=0.$$
Substituting $x=e^{-y}$ for the first two and $x=... |
3,917,283 | <p>I have the following : <span class="math-container">$f_n(x)=\frac{nx}{1+nx^3} \quad n=1,2,\dots \quad$</span> and <span class="math-container">$f(x)=\lim_{n \to \infty} f_n(x) ,$</span></p>
<p>and I have done the following : <span class="math-container">$$|f_n(x) -f(x)| = \biggl|\frac{nx}{1+nx^3}-\frac{1}{x^2}\bigg... | mechanodroid | 144,766 | <p>Assume that <span class="math-container">$f_n \to f$</span> uniformly on <span class="math-container">$\langle 0,1\rangle$</span>. For <span class="math-container">$\varepsilon = 1$</span> there exists <span class="math-container">$n_0 \in \Bbb{N}$</span> such that
<span class="math-container">$$n \ge n_0 \implies |... |
2,712,318 | <p>Let $a$ be the real root of the equation $x^3+x+1=0$</p>
<p>Calculate $$\sqrt[\leftroot{-2}\uproot{2}3]{{(3a^{2}-2a+2)(3a^{2}+2a)}}+a^{2}$$</p>
<p>The correct answer should be $ 1 $. I've tried to write $a^3$ as $-a-1$ but that didn't too much, I guess there is some trick here :s</p>
| hamam_Abdallah | 369,188 | <p>The expansion of what is under square root gives</p>
<p>$$A=9a^4+6a^3-6a^3-4a^2+6a^2+4a=$$
$$a (9a^3+2a+4)=a (9 (-a-1)+2a+4)=$$
$$-a (7a+5)=$$</p>
<p>this must be equal to
$$B=(1-a^2)^3=1-3a^2+3a^4-a^6=$$
$$$$</p>
<p>the difference is
$$B-A=1+4a^2+5a+3a^4-(a+1)^2=$$
$$3a^2+3a+3a^4=3a (a^3+a+1)=0$$</p>
<p>It is c... |
2,712,318 | <p>Let $a$ be the real root of the equation $x^3+x+1=0$</p>
<p>Calculate $$\sqrt[\leftroot{-2}\uproot{2}3]{{(3a^{2}-2a+2)(3a^{2}+2a)}}+a^{2}$$</p>
<p>The correct answer should be $ 1 $. I've tried to write $a^3$ as $-a-1$ but that didn't too much, I guess there is some trick here :s</p>
| mz71 | 301,435 | <p>Maybe there is a trick to it, but good ol' factoring can work here.</p>
<p>It is easy to check (using the identity $a^3=-a-1$) that
$$ (3a^2-2a+2)(3a^2+2a) = -5a-7a^2 $$
Let $\sqrt[3]{-5a-7a^2}+a^2 = y$. Given that $a\in \mathbb{R}$, we know that also $y\in \mathbb{R}$, and we must solve for it.
$$y-a^2 = \sqrt[3]{... |
4,432,047 | <p><span class="math-container">$3\times 2$</span> is <span class="math-container">$3+3$</span> or <span class="math-container">$2+2+2$</span>. We know both are correct as multiplication is commutative for whole numbers. But which one is <strong>mathematically</strong> accurate?</p>
| Mark Bennet | 2,906 | <p>The answer to this question depends on context. Strictly you need also addition to be associative even to know that <span class="math-container">$2+2+2$</span> is well-defined. Addition is usually taken to be a binary operation, and if you want to be very strict you should be expressing it as such.</p>
<p>Both answe... |
80,452 | <p>When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form</p>
<p>'{${E_i\wedge E_j }$}' where '{${E_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?</p>
| Thomas Richard | 8,887 | <p>I am not aware of any systematic treatment of this question. However there are two standard examples.</p>
<p>1) If $(M,g)$ is an hyper surface in a constant curvature manifold, then its curvature operator is given by $R=A\wedge A-kI$ where $I$ is the identity and $A$ is the second fundamental form. Then if $e_i$ di... |
80,452 | <p>When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form</p>
<p>'{${E_i\wedge E_j }$}' where '{${E_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?</p>
| Robert Bryant | 13,972 | <p>There are still a few interesting things to say about this question, so I thought I'd add some comments.</p>
<p>In one sense, the answer to the question of when a Riemannian metric has an orthonormal coframing that diagonalizes the curvature in the manner requested by the OP is an algebraic problem: As is well-kno... |
13,616 | <p>Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.</p>
<p>I think that the tags are relevant, but feel free to change them.</p>
<p>Also, have ther... | Marc Nieper-Wißkirchen | 1,841 | <p>A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.</p>
<p>So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in a... |
4,352,129 | <p>I am struggling to compute the power series expansion of <span class="math-container">$$f(z) = \frac{1}{2z+5}$$</span> about <span class="math-container">$z=0$</span>, where <span class="math-container">$f$</span> is a complex function. I tried comparing it to the geometric series as follows,<span class="math-contai... | Riemann'sPointyNose | 794,524 | <p>You've gone wrong because
<span class="math-container">$$
\frac{1}{1-\omega}=\sum_{n\geq 0}\omega^n
$$</span>
assumes that <span class="math-container">${|\omega|<1}$</span>. In your case, this would mean <span class="math-container">${|2z+4|<1}$</span>, i.e. <span class="math-container">${|z+2|<\frac{1}{2}... |
2,433,614 | <p>The solution to the initial-value problem $y' = y^{2} + 1$ with $y(0)=1$ is $y = \tan(x + \frac{\pi}{4})$. I would like to show that this is the correct solution in a way that is analogous to my solution to differential equation $y' = y - 12$.</p>
<p><strong>Solution to Differential Equation</strong></p>
<p>If $y'... | anomaly | 156,999 | <p>Start out by defining $\sin$ and $\cos$ to be the unique solutions to the differential equation $y'' = -y$ with the boundary conditions $y(0) = 0, y'(0) = 1$ and $y(0) = 1, y'(0) = 0$, respectively. (It's not difficult to show that such functions exist and are unique, and one can verify that the usual Taylor series ... |
927,815 | <p>Find an equation of the plane that passes through the points $A(0, 1, 0)$, $B(1, 0, 0)$ and $C(0, 0, 1)$.</p>
| Novice | 97,093 | <p>Apologies. I view the question wrongly.</p>
<p>Find the vectors $\vec{AB}$ and $\vec{BC}$:</p>
<p>$\vec{AB}=\left(
\begin{array}{c}
1\\
-1\\
0\\
\end{array}
\right)$</p>
<p>$\vec{BC}=\left(
\begin{array}{c}
1\\
0\\
-1\\
\end{array}
\right)$</p>
<p>Take the cross product and it will be the normal of the plane.</p... |
3,827,853 | <p>I am trying to compute the singular cohomology of <span class="math-container">$\mathbb{N}\subset\mathbb{R}$</span>. I have that <span class="math-container">$C_0(\mathbb{N})\cong C_1(\mathbb{N})\cong\mathbb{Z}^{\oplus\mathbb{N}}$</span>, where I mean <span class="math-container">$\mathbb{Z}^{\oplus\mathbb{N}}=\{(n_... | Arturo Magidin | 742 | <p>This gives the same answer as the one posted, but invoking categorical arguments.</p>
<p>Since the direct sum is the coproduct, and maps <em>from</em> a coproduct correspond to families of maps from the constituents, each morphism <span class="math-container">$f\colon \oplus_{i\in I}A_i\to B$</span> corresponds to a... |
975,207 | <p><strong>Question:</strong></p>
<blockquote>
<p>Solve the following system for $a,b,c\in \mathbb{R}$:
$$\begin{cases}
b^2-6=2\sqrt{2a+6}\\
c^2-6=2\sqrt{2b+6}\\
a^2-6=2\sqrt{2c+6}
\end{cases}$$</p>
</blockquote>
<p>I found the following:$$ (b^2-6)^2=4(2a+6)$$
$$(c^2-6)^2=4(2b+6)$$
$$(a^2-6)^2=4(2c+6)$$
Then mayb... | HK Lee | 37,116 | <p>Let $$ 3T:=a^2+b^2+c^2-18 = 2\sqrt{2a+6} +
2\sqrt{2b+6} + 2\sqrt{2c+6} $$</p>
<p>If $$ a= {\rm max}\ \{ a,b,c\},\ b
= {\rm min}\ \{ a,b,c\},\ a>b$$ then
$$T>b^2-6=2\sqrt{2a+6} > T$$</p>
<p>Contradiction.</p>
|
975,207 | <p><strong>Question:</strong></p>
<blockquote>
<p>Solve the following system for $a,b,c\in \mathbb{R}$:
$$\begin{cases}
b^2-6=2\sqrt{2a+6}\\
c^2-6=2\sqrt{2b+6}\\
a^2-6=2\sqrt{2c+6}
\end{cases}$$</p>
</blockquote>
<p>I found the following:$$ (b^2-6)^2=4(2a+6)$$
$$(c^2-6)^2=4(2b+6)$$
$$(a^2-6)^2=4(2c+6)$$
Then mayb... | Arian | 172,588 | <p>$$(b^2-6)^2-(c^2-6)^2=8(a-b)\Leftrightarrow (b^2-c^2)(b^2+c^2-12)=8(a-b)$$
$$(b^2-6)^2-(a^2-6)^2=8(a-c)\Leftrightarrow (b^2-a^2)(b^2+a^2-12)=8(a-c)$$
$$(c^2-6)^2-(a^2-6)^2=8(b-c)\Leftrightarrow (c^2-a^2)(c^2+a^2-12)=8(b-c)$$
Assume that $a\geq b\geq c\geq 0$ then
$$b^2+c^2\geq 12$$
on the other hand
$$b^2+a^2\leq12... |
515,915 | <p><strong>Definiton</strong> A $p$-adic integer is a (formal) series
$$\alpha=a_0+a_1p+a_2p^2+\ldots$$
with $0\leq a_i<p$.</p>
<p>The set of $p$-adic integers is denoted by $\mathbb{Z}_p$. If we cut an element $\alpha\in\mathbb{Z}_p$ at its $k$-th term
$$\alpha_k=a_0+a_1p+\ldots+\alpha_{k-1}p^{k-1}$$
$\textbf{we g... | dfeuer | 17,596 | <p>This holds only in the plane, not in 3-space. Do you see why?</p>
<p><strong>Finding an argument:</strong> In the complex plane, a rotation is multiplication by $e^{i\theta}$ and translation is addition of a constant. So rotation followed by translation is $f(x)=e^{i\theta}x+c$ while translation followed by rotatio... |
2,447,014 | <p>I attempted to solve this question using a method learned from a previous answer on here. I was just looking for a bit more guidance.
This is what I have for this problem:</p>
<blockquote>
<p>$y=\frac{4x}{x+1}\\
y(x+1) =4x \\
yx + y = 4x \\
y = 4x-yx \\
y= x(4-y) \\
\frac{y}{4-y}=x \\
\frac{y}{4-y} > 0 \\
0 &... | marty cohen | 13,079 | <p>$\frac{4x}{x+1}
=\frac{4x+4-4}{x+1}
=\frac{4(x+1)-4}{x+1}
=4-\frac{4}{x+1}
$</p>
<p>Now, the ranges.</p>
<p>$\begin{array}\\
x+1:
&(0, \infty)\to (1, \infty)\\
\frac1{x}:
&(1, \infty) \to (0, 1)\\
4x:
&(0, 1)\to (0, 4)\\
4-x:
&(0, 4) \to 0, 4)\\
\end{array}
$
So the range is
$(0, 4)$.</p>
|
265,549 | <p>$$ \left\{ x \in\mathbb{R}\; \middle\vert\; \tfrac{x}{|x| + 1} < \tfrac{1}{3} \right\}$$</p>
<p>What is the supremum and infimum of this set? I thought the supremum is $\frac{1}{3}$. But can we say that for any set $ x < n$ that $n$ is the supremum of the set? And for the infimum I have no idea at all. Also,... | M. Strochyk | 40,362 | <p>Considering separately cases $x\geqslant{0}$ and $x<0$ you can find range of $x$ for which $f(x)=\dfrac{x}{|x| + 1} - \dfrac{1}{3} <0.$</p>
<p>For $x\geqslant {0}$ $$f(x)=\dfrac{x}{x+1}-\dfrac{1}{3}=\dfrac{x+1-1}{x+1}-\dfrac{1}{3}=\dfrac{2}{3}- \dfrac{1}{x+1},$$
For $x<0$ $$f(x)=\dfrac{x}{1-x}-\dfrac{1}{3}... |
2,873,449 | <p>For a continuous function $f: R \to R $ Define:
$$ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t)t \space dt $$
Since the function is continuous I can assume that it's also integrable since continuity implies integrability. I assume furthermore that there exists a function $F$ which is an antiderivative of $f$ for w... | Ted Shifrin | 71,348 | <p>By L'Hôpital's rule, this limit is equal to
$$\lim_{x\to 0} \frac{xf(x)}{2x} = \frac12\lim_{x\to 0} f(x) = \frac{f(0)}2.$$
(Use the first Fundamental Theorem of Calculus to differentiate the integral, since the integrand is guaranteed continuous.) You were correct.</p>
|
1,797,245 | <p>How does one show that $I_n = \int\limits_0^1 x^n e^x dx$ is decreasing?</p>
<p>The best I came up with is this: $I_{n-1} - I_n= \int\limits_0^1 e^xx^{n-1}(1-x)dx$, but how do we go from here?</p>
<p>I'd appreciate some hints.</p>
| rtybase | 22,583 | <p>According to <a href="https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_Value_Theorems_for_Definite_Integrals" rel="nofollow">Mean Value Theorem</a>, there $\exists c \in (0,1)$ such that
$$\int_{0}^{1}e^xx^{n-1}(1-x)dx=e^cc^{n-1}(1-c)(1-0)\geq0$$</p>
<p>because $f(x)=e^xx^{n-1}(1-x)$ is continuous on $[0,1]$<... |
1,237,464 | <p>In a bag of reds and black balls, $30\%$ were red, and $90\%$ of the black balls and $80\%$ of the red balls are marked balls. What percentage of the bag of balls is marked?</p>
<p>I thought I would have to use: Let </p>
<p>$A = \text{marked red balls},$</p>
<p>$B = \text{marked black balls}.$ </p>
<p>$P(A \cup ... | Community | -1 | <p>Let in bag be $x$ balls.<br/>
Number of red balls: $30\%x$<br/>
Number of black balls: $(100-30)\%x=70\%x$<br/>
Number of marked red balls: $80\%\cdot30\%x=24\%x$<br/>
Number of marked black balls: $90\%\cdot70\%x=63\%x$<br/>
So, answer will be: $24\%x+63\%x=87\%x$</p>
|
1,237,464 | <p>In a bag of reds and black balls, $30\%$ were red, and $90\%$ of the black balls and $80\%$ of the red balls are marked balls. What percentage of the bag of balls is marked?</p>
<p>I thought I would have to use: Let </p>
<p>$A = \text{marked red balls},$</p>
<p>$B = \text{marked black balls}.$ </p>
<p>$P(A \cup ... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>$P\left(\text{marked}\right)=P\left(\text{marked and black}\right)+P\left(\text{marked and red}\right)$</p>
<p>$P\left(\text{marked}\right)=P\left(\text{marked}\mid\text{black}\right)P\left(\text{black}\right)+P\left(\text{marked}\mid\text{red}\right)P\left(\text{red}\right)$</p>
<hr... |
1,405,787 | <p>We're given the following function :</p>
<p>$$f(x,y)=\dfrac{1}{1+x-y}$$</p>
<p>Now , how to prove that the given function is differentiable at $(0,0)$ ?</p>
<p>I found out the partial derivatives as $f_x(0,0)=(-1)$ and $f_y(0,0)=1$ ,</p>
<p>Clearly the partial derivatives are continuous , but that doesn't guaran... | Jack | 206,560 | <p>You need to show that the limit definition of the partial derivative from each direction is the same value. Hence</p>
<p>$\frac{\partial f}{\partial x}(0,0) = \lim_{h \to 0^{-}} \frac{f(0+h,0)-f(0,0)}{h} = \lim_{h \to 0^{-}} \frac{\frac{1}{1+h} - 1}{h} = \lim_{h \to 0^{-}} -\frac{1}{(1+h)^{2}} = -1$ (Using l'hopit... |
1,628,386 | <p>I had following limit of two variables as a problem on my calculus test. How does one show whether the limit below exists or does not exist? I think it does not exist but I was not able to show that rigorously. There was a hint reminding that $\lim_{t\to 0}\sin t / t=1$.</p>
<p>$$\lim_{(x,y)\to (0,0)} \frac{3x^2\si... | marty cohen | 13,079 | <p>The first approximation
I would use is
$\sin(x)
\approx x$
for small $x$.</p>
<p>Therefore
$\frac{3x^2\sin^2y}{2x^4+2\sin y^4}
\approx \frac{3x^2y^2}{2x^4+2y^4}
= \frac{3(x/y)^2}{2(x/y)^4+2}
$
(dividing by $y^4$).</p>
<p>Therefore,
if $x/y=c$,
this approaches
$\frac{3c^2}{2c^4+2}
$,
which can take on a variety of ... |
1,201,904 | <p>I have to implement a circuit following the boolean equation A XOR B XOR C, however the XOR gates I am using only have two inputs (I am using the 7486 XOR gate to be exact, in case that makes a difference)... is there a way around this?</p>
| copper.hat | 27,978 | <p>Exclusive or $\oplus$ is associative (and commutative), so you can use any order
and any pairing.</p>
<p>$A \oplus B \oplus C = (A \oplus B) \oplus C = A \oplus (B \oplus C) $.</p>
|
82,061 | <p>Well, another problem here I don't get.</p>
<p>From what I know, sin(x)'=cos(x) right?</p>
<p>Well, here is a problem:</p>
<p>Find $\frac{d}{dx}(y\cos(\frac{y}{x^4}))$ ?</p>
<p>Let $u=y\cos(\frac{y}{x^4})$ and $s=y$, $v=\cos(\frac{y}{x^4})$</p>
<p>$$u'=v\frac{ds}{dx}+s \frac{dv}{dx}$$</p>
<p>$$u'=\co... | Ross Millikan | 1,827 | <p>For $k=2$ you can use the fact that $\left(\sum_n a_n\right)^2=\left(\sum_n a_n^2\right)+2\sum_{i<j}a_ia_j$. Your expression is the third term (without the factor $2$). For $k=3,$</p>
<p>$ \left(\sum_n a_n\right)^3=\left(\sum_n a_n^3\right)+3\sum_{i\ne j}a_i^2a_j+6\sum_{i<j<k}a_ia_ja_k$</p>
<p>$=\left(\... |
2,266,342 | <p>So on my review for my final exam there is this question: </p>
<p>Is there a linear transformation from $P_2$ to $P_2$ with the following properties? In each case, either give an example of such a transformation or prove that no such transformation exist.</p>
<blockquote>
<p>$T(t^2+t+1) = t^2+t+1, T(t^2+2t+3) = ... | Robert Israel | 8,508 | <p>Are $t^2 + t + 1$, $t^2 + 2 t + 3$, $t^2 + 2 t + 2$ linearly independent? If so, then they form a basis and there is such a linear transformation. If not, you must determine whether these equations are consistent.</p>
|
2,902,768 | <p>$f:\mathbb{R}^2 \to \mathbb{R}$</p>
<p>$f\Bigg(\begin{matrix}x\\y\end{matrix}\Bigg)=\begin{cases}\frac{xy^2}{x^2+y^2},(x,y)^T \neq(0,0)^T \\0 , (x,y)^T=(0,0)^T\end{cases}$</p>
<p>I need to determine all partial derivatives for $(x,y)^T \in \mathbb{R}^2$:</p>
<p>$f_x=y^2/(x^2+y^2)-2x^2y^2/(x^2+y^2)^2$ for $(x,y)^T... | mechanodroid | 144,766 | <p>The partial derivatives at $(0,0)$ are</p>
<p>$$\partial_x f(0,0) = \lim_{h\to 0} \frac{f(h,0) - f(0,0)}h = 0$$
$$\partial_y f(0,0) = \lim_{h\to 0} \frac{f(0,h) - f(0,0)}h = 0$$</p>
<p>so the only candidate for the differential $Df(0,0)$ is the zero operator.</p>
<p>However, the limit</p>
<p>$$\lim_{(h_1, h_2) \... |
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| user26486 | 107,671 | <p>The other answers here show the strategies of:</p>
<hr>
<p>$1$) Substitution (the most general). See Kim Jong Un's answer on this question.</p>
<hr>
<p>$2)$ Solving a quadratic equation that has the roots $x,y$ (this uses the <a href="http://en.wikipedia.org/wiki/Vieta's_formulas" rel="nofollow">Vieta's form... |
339,423 | <p>Assuming $\lbrace R_i \rbrace _{i=1}^{n}$ be a set of rings, show that $$e=(e_1,...,e_n)$$ is the identity of $$R=R_1 \times R_2 \times ... \times R_n$$ if and only if $e_i$ is the identity of $R_i$ for all $1 \leq i \leq n$. I come across this statement when $R$ is the internal direct product of $R_i$ but here, we ... | Branimir Ćaćić | 49,610 | <p>$
\newcommand{\so}{\mathfrak{sl}}
\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\bC}{\mathbb{C}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\tr}{tr}
\newcommand{\i}[1]{{#1}^{-1}}
\newcommand{\c}[2]{{#2}{#1}\i{{#2}}}
$Let $\so(2,\bC)$ denote the comple... |
339,423 | <p>Assuming $\lbrace R_i \rbrace _{i=1}^{n}$ be a set of rings, show that $$e=(e_1,...,e_n)$$ is the identity of $$R=R_1 \times R_2 \times ... \times R_n$$ if and only if $e_i$ is the identity of $R_i$ for all $1 \leq i \leq n$. I come across this statement when $R$ is the internal direct product of $R_i$ but here, we ... | Ted | 15,012 | <p><span class="math-container">$\DeclareMathOperator{\tr}{tr}$</span>The map <span class="math-container">$(A,B) \mapsto \tr(AB)$</span> defines a symmetric, nondegenerate bilinear form on the space of traceless matrices in <span class="math-container">$M_2\mathbb{C}$</span>. Clearly the action of <span class="math-c... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>$$\sum\limits_{n=1}^{\infty} n = 1 + 2 + 3 + \cdots \text{ad inf.} = -\frac{1}{12}$$</p>
<p>You can also see many more here: <a href="http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/" rel="nofollow">The Euler-Maclaurin ... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user02138 | 2,720 | <p>\begin{eqnarray}
\sum_{k = 0}^{\lfloor q - q/p) \rfloor} \left \lfloor \frac{p(q - k)}{q} \right \rfloor = \sum_{k = 1}^{q} \left \lfloor \frac{kp}{q} \right \rfloor
\end{eqnarray}</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>Ah, this is one identity which comes into use for proving the <em>Euler's Partition Theorem</em>. The identity is as follows: $$ (1+x)(1+x^{2})(1+x^{3}) \cdots = \frac{1}{(1-x)(1-x^{3})(1-x^{5}) \cdots}$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Yrogirg | 14,862 | <p>I actually think <a href="http://en.wikipedia.org/wiki/Currying">currying</a> is really cool:</p>
<p>$$(A \times B) \to C \; \simeq \; A \to (B \to C)$$</p>
<p>Though not strictly an identity, but an isomorphism.</p>
<p>When I met it for the first time it seemed to be a bit odd but it is so convenient and neat. A... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Z. L. | 55,611 | <p>$32768=(3-2+7)^6 / 8$</p>
<p>Just a funny coincidence.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| marty cohen | 13,079 | <p>$\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3) = \pi$
(using the principal value),
but if you blindly use the addition formula
$\tan^{-1}(x) + \tan^{-1}(y)
= \tan^{-1}\dfrac{x+y}{1-x y}$
twice, you get zero:</p>
<p>$\tan^{-1}(1) + \tan^{-1}(2) =
\tan^{-1}\dfrac{1+2}{1-1*2}
=\tan^{-1}(-3)$;
$\tan^{-1}(1) + \tan^{-1}(2) + \... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| xpaul | 66,420 | <p>I have one: In a $\Delta ABC$,
$$\tan A+\tan B+\tan C=\tan A\tan B\tan C.$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Felix Marin | 85,343 | <p>$${\Large%
\sqrt{\,\vphantom{\huge A}\color{#00f}{20}\color{#c00000}{25}\,}\, =\ \color{#00f}{20}\ +\ \color{#c00000}{25}\ =\ 45}
$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Jaycob Coleman | 76,533 | <p>Let $\sigma(n)$ denote the sum of the divisors of $n$.</p>
<p>If $$p=1+\sigma(k),$$ then $$p^a=1+\sigma(kp^{a-1})$$
where $a,k$ are positive integers and $p$ is a prime such that $p\not\mid k$.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| ZHN | 131,755 | <p>$$\frac{\pi}{4}=\sum_{n=1}^{\infty}\arctan\frac{1}{f_{2n+1}}, $$
where $f_{2n+1}$ there are fibonacci numbers, $n=1,2,...$</p>
|
4,556,592 | <p>Check if the applications defined below are linear transformations:</p>
<p>a) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-container">$T(x_1, x_2) = (x_1 – 1, x_2).$</span></p>
<p>b) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-con... | 311411 | 688,046 | <p><em>Partial Answer:</em></p>
<p>For part (a) you are not applying the rule of function <span class="math-container">$T$</span> correctly.</p>
<p><span class="math-container">$$(a+)\quad T(u_1+v_1, u_2+v_2) = (u_1 + v_1 – 1, u_2 + v_2) \not = (u_1 + v_1 – 2, u_2 + v_2).$$</span></p>
<p><span class="math-container">... |
4,556,592 | <p>Check if the applications defined below are linear transformations:</p>
<p>a) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-container">$T(x_1, x_2) = (x_1 – 1, x_2).$</span></p>
<p>b) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-con... | Shubham Namdeo | 206,275 | <p>a). Note that <span class="math-container">$T(0,0)=(-1,0)\neq (0,0)$</span>. Therefore, <span class="math-container">$T$</span> fails to be linear map. Because if <span class="math-container">$T$</span> was linear then <span class="math-container">$T(0,0)=T(-1 \cdot (0,0)) = -1 T(0,0)$</span> and this implies that <... |
3,687,484 | <p>Here, <span class="math-container">$H^1(\mathbb{R}^2)$</span> is the standard Sobolev spaces for <span class="math-container">$L^2(\mathbb{R}^2)$</span> functions whose weak derivative belongs to <span class="math-container">$L^2(\mathbb{R}^2).$</span></p>
<p>My question in the title comes from calculus of variatio... | Yung-Hsiang Huang | 225,984 | <p>This seems to be <strong>not</strong> a general property for functions in <span class="math-container">$H^1(\mathbb{R}^2) \cap C(\mathbb{R}^2)$</span> because I find the following remark from <a href="https://www.springer.com/gp/book/9780857292261" rel="nofollow noreferrer">this book</a>.</p>
<p><a href="https://i.... |
495,069 | <p>Find an equation of the plane.
The plane that passes through the line of intersection of the planes
$x − z = 1$ and $y + 4z = 1$
and is perpendicular to the plane
$x + y − 2z = 2$.</p>
<p>I keep getting the answer of $7x-y+5z=6$ and I am told that it is wrong. I do not understand what I am doing wrong.
I have th... | Cameron Buie | 28,900 | <p>$$\langle 1,-4,1\rangle\times\langle 1,1,-2\rangle=\langle 7,3,5\rangle$$</p>
|
1,310,530 | <p>From: $2015$ Singapore Mathematical Olympiad Secondary 2 (Grade 8) Question 21 Round 1 on 3rd June.</p>
<blockquote>
<p>Find the value of $\sqrt {(98 \times 100+2)(100\times102+2)+(100\times2)^2}$ (No use of calculators)</p>
</blockquote>
<p>My attempt:Special cases formula -> $x^2=(x+1)(x-1)+1$</p>
<p>Therefor... | mathlove | 78,967 | <p>Note that $$98\times 100+2\not =99^2-1$$
and that
$$100\times 102+2\not=101^2-1.$$</p>
<p>We have
$$98\times 100+2=(99-1)(99+1)+2=99^2-1+2=99^2+1$$
and
$$100\times 102+2=(101-1)(101+1)+2=101^2-1+2=101^2+1.$$</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | roy smith | 9,449 | <p>If it interests you.</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Ronnie Brown | 19,949 | <p>Dan Schechtman, winner of the 2011 Nobel Prize in Chemistry for the discovery of quasi crystals, said: “The main lesson that I have learned over time is that a good scientist is a humble and listening scientist and not one that is sure 100 percent in what [they read] in the textbooks.”</p>
<p>My research on groupoi... |
3,843,559 | <p>Can someone please give me a hint why for metric spaces we have</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y)<\varepsilon\}\subset \{x|d_1(x,y)<\varepsilon\}$</span></p>
<p>I have expected the opposite:</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y... | Gabriel Delgado | 671,176 | <p>The way I see it is a matter of precision, think that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are points and <span class="math-container">$d_1$</span> is the distance in centimeters and <span class="math-container">$d$</span> is in millimeters, so numerically we have <span... |
1,729,893 | <p>Let $\nu_1, \nu_2, \nu_3 \in \mathbb{R}$ not all be zero.</p>
<p>I wish to show
$$\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0\text{.}$$</p>
<p><a href="http://www.wolframalpha.com/input/?i=x%5E2%2Bx*y%2By%5E2%2By*z%2Bz%5E2+%3E+0" rel="nofollow">Wolfram</a> seems to suggest splitting this into cases,... | marty cohen | 13,079 | <p>Expanding Deathkamp Drone's suggestion,
though I thought of it independently:</p>
<p>We want to show that
$a^2+ab+b^2+bc+c^2 > 0
$.</p>
<p>This is the same as
$2a^2+2ab+2b^2+2bc+2c^2 > 0
$.</p>
<p>$\begin{array}\\
2a^2+2ab+2b^2+2bc+2c^2
&=a^2+a^2+2ab+b^2+b^2+2bc+c^2+c^2\\
&=a^2+(a+b)^2+(b+c)^2+c^2\\... |
356,940 | <p>According to Wolfie:</p>
<p>$2^{-1} \bmod 5 = 3$</p>
<p><a href="http://www.wolframalpha.com/input/?i=2%5E-1+mod+5" rel="nofollow">http://www.wolframalpha.com/input/?i=2%5E-1+mod+5</a></p>
<p>Why is that?</p>
| HSN | 58,629 | <p>You need to think by yourself what inverses mean, without thinking about mod 5 first. The multiplicative inverse of $a$ -if it exists- is some element $b$ such that $a\cdot b = 1$.</p>
<p>Now work mod 5. The inverse of $2$ is an element $b\in\mathbb{Z}/(5)$ such that 2*b=1. It shouldn't be too hard to verify that $... |
1,843,724 | <p>I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$
But i'm not sure how to find the Maclaurin series here
I tried this</p>
<p>$$
f'_{(0)}=-2xe^{-x^2}=0
$$
And that follows to every derivative that follows, so how can I get a power series out of it?</p>
| Enrico M. | 266,764 | <p>That is easier than what you think.</p>
<p>General expansion:</p>
<p>$$e^A = 1 + A + \frac{1}{2}A^2 + \frac{1}{6}A^3 + \cdots$$</p>
<p>So if $A = -x^2$ you get</p>
<p>$$e^{-x^2} = 1 - x^2 + \frac{1}{2}(-x^2)^2 + \frac{1}{6}(-x^2)^3 + \cdots$$</p>
<p>So</p>
<p>$$e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{... |
2,086,134 | <p>Solve for $x$ if $4^x + x^x = 20$ where $x \in \mathbb{Z}$ </p>
<p>My Attempt: </p>
<p>By inspection,<br>
$x=2$</p>
<p>Now,
$4^2 + 2^2 = 20$</p>
<p>$16+4=20$</p>
<p>$20=20$</p>
<p>Which is True.</p>
<p>But, I Could not do it by any process. Can anyone help me? </p>
<p>Thanks in Advance.</p>
| Community | -1 | <p>If the solution is an integer,</p>
<p>$$4^1+1^1<20,\\4^0+0^0<20^*,\\\frac1{4^n}+\frac{(-1)^n}{n^n}\le1+1<20$$ and for all $n>2$, $$4^n+n^n\ge16+4=20.$$</p>
<hr>
<p>$^*$Whatever the meaning of $0^0$.</p>
|
341,621 | <p>Is there an inequality such as
$$(a+b)^2 \leq 2(a^2 + b^2)$$
for higher powers of $k$
$$(a+b)^k \leq C(a^k + b^k)?$$</p>
| Giuseppe Negro | 8,157 | <p>Take the ratio $$f(a, b)=\frac{(a+b)^k}{a^k+b^k}$$and observe that it is homogeneous of degree zero, that is $f(a, b)=f(\lambda a, \lambda b)$ for all $\lambda >0$. So $f$ is constant along rays in $\mathbb{R}^2$ and so, in particular,
$$f(a, b)\le \max_{(x, y)\in \mathbb{S}^1} f(x, y),$$
where $\mathbb{S}^1$ is... |
2,703,559 | <p>My assignment asks me to prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$. I have been stuck for ages, and now I wonder if it is true. I need some help to get started. This is my attempt: Assume $k$ and $q$ are relative prime (this is necessary other... | cansomeonehelpmeout | 413,677 | <p>You are almost there. </p>
<p>You're right that $k=-1$ is a possibility, since $k^2\equiv_q 1$ (you also know that there are atmost two values $k$ can have, since $\mathbb{Z}_q$ is a field). Remember that if $k=1$, i.e, $m\mapsto m$ then the order of the automorphism is $1$, not $2$.</p>
|
3,892,595 | <p>While solving number theory problems I sometimes I have to use a function that can be defined as</p>
<p><span class="math-container">$$
f(n) =
\begin{cases}
1 & \text{ if } n = 1, \\
0 & \text{ if } n > 1.
\end{cases}
$$</span></p>
<p>where <span class="math-container">$ n $</span> is a positive integer.<... | davidlowryduda | 9,754 | <p>In Apostol's Introduction to Analytic Number Theory, he calls this function <span class="math-container">$I$</span>.</p>
<p><a href="https://i.stack.imgur.com/Voa4A.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Voa4A.png" alt="enter image description here" /></a></p>
<p>Part of Apostol's rationa... |
3,892,595 | <p>While solving number theory problems I sometimes I have to use a function that can be defined as</p>
<p><span class="math-container">$$
f(n) =
\begin{cases}
1 & \text{ if } n = 1, \\
0 & \text{ if } n > 1.
\end{cases}
$$</span></p>
<p>where <span class="math-container">$ n $</span> is a positive integer.<... | unobservable_node | 442,550 | <p>It is usually called an indicator function in signal processing. It is denoted as either <span class="math-container">$\delta(1)$</span> or <span class="math-container">$\mathbb{1}_{n=1}$</span>.</p>
|
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | apnorton | 23,353 | <p><strong>Spoiler Alert:</strong> <em>(I use the answer given above in the response below. If you don't want to see it, you may want to skip this answer...)</em></p>
<p>I'm replacing $\times$ by $\circ$, as the latter is more commonly used with unknown operations. I hate it when people redefine a common symbol, th... |
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | haqnatural | 247,767 | <p>You can prove it as follows:</p>
<p>for every $\varepsilon >0$ and $m+1>\left| a \right| $ and if $n$ is big enough then
$$0<\left| \frac {a^n}{ n! } \right| =\frac { \left| a \right| }{ 1 } \cdot \frac { \left| a \right| }{ 2 } \cdots \frac { \left| a \right| }{ m } \cdot \frac { \left| a \right| ... |
163,043 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/160847/polynomials-irreducible-over-mathbbq-but-reducible-over-mathbbf-p-for">Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$</a> </p>
</blockquote>
<p>Anyon... | Community | -1 | <p>"Most" polynomials are irreducible. So if we can find any polynomial reducible modulo 2, 3, and 5, we should have no trouble finding one that is also irreducible over the integers.</p>
<p>And we can even consider the three primes independently, and use the Chinese Remainder Theorem (CRT) to combine our results.</p... |
2,172,870 | <p>I have two vectors $u$, and $v$, I know that $\mid u \mid$ = 3 and $\mid v \mid$ = 5, and that $u\cdot v = -12$. I need to calculate the length of the vector $(3u+2v) \times (3v-u)$.</p>
<p>Because I know the dot product of the vectors I know what cosine of the angle between them is $\cos \theta = -0.8$, and also $... | Lelouch | 152,626 | <p>The definition of a 'closed' knights tour on a $m \times n$board, is a sequence of steps from a starting square $a_1$ to another square $a_{mn}$ , such that every square is visited exactly once, and the last sqaure is only one knight step away from $a_1$. Having said that, it is obvious, that for $mn $(mod2) $= 1$, ... |
3,609,791 | <p><span class="math-container">$f(x) = \frac{x}{(1+x^2)}$</span></p>
<p>I need to find the range of function.</p>
<p>My method-</p>
<p>Let <span class="math-container">$f(x)=y$</span> </p>
<p>Then <span class="math-container">$x²y-x+y=0$</span></p>
<p><span class="math-container">$$x=1\pm\frac{\sqrt{(1-4y²)}}{2y}... | Michael Rozenberg | 190,319 | <p>This can be equal to <span class="math-container">$0$</span>.</p>
<p>You solved this problem for <span class="math-container">$y\neq0$</span>.</p>
<p>Now, easy to see that <span class="math-container">$y=0$</span> holds.</p>
<p>I like the following way.</p>
<p>Let <span class="math-container">$x>0.$</span></p... |
4,356,457 | <p>Let <span class="math-container">$k$</span> be a finite field of size <span class="math-container">$q$</span>, let <span class="math-container">$n\ge1$</span>, and let <span class="math-container">$\mathrm{GL}_n(k)$</span> and <span class="math-container">$M_n(k)$</span> be the group of invertible matrices and ring ... | reuns | 276,986 | <p>I find <span class="math-container">$$\sum_{n\ge 0} a_n t^n=\prod_{e=1}^\infty \frac1{1-t^e|k|} \qquad \text{and}\qquad\sum_{n\ge 0} b_n t^n=\prod_{e=1}^\infty \frac{1-t^e}{1-t^e|k|}$$</span> for the generating functions of the number of <span class="math-container">$GL_n(k)$</span>-conjugacy classes in <span class=... |
264,595 | <p>I've been trying to find an asymptotic expansion of the following series</p>
<p>$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$</p>
<p>and</p>
<p>$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$</p>
<p>around $+\infty$, in the from</p>
<p>$$\exp(x^2)\Big(1+\frac{a_1}{... | esg | 48,831 | <p>I sketch the arguments for $C(x)$, the arguments for $L(x)$ are essentially the same.</p>
<p>The specific form of the sum suggests probabilistic arguments.
Let $X_x$ be a $\mathrm{Poiss}(x^2)$-distributed random variable and note that
$$C(x)\,e^{-x^2}=\mathbb{E}\frac{x}{\sqrt{X_x}}\,1_{\{X_x\geq 1\}}$$
It is know... |
3,636,040 | <p>I am reading a paper titled "Cross-Calibration for Data Fusion of EO-1/Hyperion and Terra/ASTER" which mentions a particular minimization problem
<img src="https://i.stack.imgur.com/UgZrB.png" alt="Frobenius norm of y(I)-r(I)x"></p>
<p>I am not able to comprehend how this equation is converted (during implementatio... | Siong Thye Goh | 306,553 | <p>For your paper <span class="math-container">$\tilde{y}_i$</span> and <span class="math-container">$r_i$</span> are row vectors. </p>
<p>We have </p>
<p><span class="math-container">\begin{align}\|\tilde{y_i}-r_iX_i\|_F^2 &= (\tilde{y_i}-r_iX_i)(\tilde{y_i}-r_iX_i)^T\\
&=\|\tilde{y_i}\|_F^2-2r_iX_i\tilde{y}... |
3,169,258 | <p>I need to evaluate on the circle <span class="math-container">$\left|z\right|=\pi$</span> the integral
<span class="math-container">$$\int_{\left|z\right|=\pi}\frac{\left|z\right|e^{-\left|z\right|}}{z}dz.$$</span>
The function is not holomorphic there. Anyway, I tried to integrate it using polar coordinates and sim... | Martín-Blas Pérez Pinilla | 98,199 | <p>Let be <span class="math-container">$z = \pi e^{i\theta}, \theta\in[0,2\pi]$</span>:
<span class="math-container">$$
\int_{|z|=\pi}\frac{|z|e^{-|z|}}{z}dz =
\int_0^{2\pi}\frac{\pi e^{-\pi}}{\pi e^{i\theta}}\pi i e^{i\theta} = 2\pi^2 i e^{-\pi}.
$$</span>
<strong>But...</strong> Cauchy formula can be used:
<span clas... |
470,081 | <p>There is a question from an old topology prelim that is somewhat giving me a hard time. Consider the cylinder $X= S^1 \times [-1,1]$. Now we define an equivalence relation $\sim$ as follows: For points $v,v' \in S^1$, we have $(v,-1) \sim (v',-1)$ and $(v,1) \sim (v',1)$. I am asked to show that the quotient space $... | Cheerful Parsnip | 2,941 | <p>Here's an alternative approach which may or may not be what your topology prelim was asking for. The quotient space is gotten by crushing each end of the cylinder $S^1\times[-1,1]$ to a point. I claim this is a 2-manifold. This is clear away from the two crushed points. Let $p$ be the point which is gotten by crushi... |
3,804,195 | <p>I am currently reworking the following exercise that was worked during my previous class's lecture on normed vector spaces:</p>
<blockquote>
<p>Show that <span class="math-container">$||x||_{\infty} \leq ||x||_{2} \leq ||x||_{1}$</span> for any <span class="math-container">$x \in \mathbb{R}^n$</span>.</p>
</blockquo... | glS | 173,147 | <p>As per <a href="https://math.stackexchange.com/a/3804250/173147">the other answer</a>, linear programming achieves this. The idea is that the problem of finding the coefficients <span class="math-container">$p_i\ge0$</span> such that <span class="math-container">$\sum_{i=1}^n p_i \mathbf x_i = \mathbf x\in\mathbb R^... |
2,131,224 | <blockquote>
<p>Say I have vectors $x, y$, then is $\text{proj }_x y $ a scalar multiple of $x$?</p>
</blockquote>
<p>I have a book saying that it is, but I have no clue why this true. Is this really true?</p>
| C. Falcon | 285,416 | <p>By definition, $\textrm{proj}_xy\in\textrm{span}(x)$, so <strong>yes</strong>, this is true.</p>
|
1,465,755 | <p>How do I find the angles $\alpha ,\beta ,\theta $ between the vector $(1,0,-1)$ and the unit vectors $i,j,k$ along the axes?</p>
<p>This question is not making sense to me. I know that in order to find the angle between any two nonzero vectors, I just have to take their dot product and divide it by the product of t... | Chappers | 221,811 | <p>Hint: $i=(1,0,0)$, $j=(0,1,0)$, $k=(0,0,1)$. Or equally, $$ (1,0,-1) = i-k $$.</p>
|
2,375,714 | <p>I know this is a really easy question but for some reason I'm having trouble with it.</p>
<p>If $M$ is an object in an additive category $\mathcal C$, and $\text{Hom}_{\mathcal C}(M,M) = 0$, then $M = 0$.</p>
<p>I know that this implies $\text{id}_M =0$ but I'm having trouble showing that $M$ satisfies the conditi... | Angina Seng | 436,618 | <p>Given an object $A$ there will be a morphism from $A$ to $M$, the
zero morphism. Composing an arbitrary morphism $f:A\to M$ with the
unique map from $M$ to $M$ you get $f$, as the unique map from $M$ to $M$
is the identity, and also the zero morphism, as it factors through the
zero object. There is therefore one mor... |
3,243,243 | <p>I am learning category theory using Basic Category Theory by Tom Leinster as my main source. In the chapter on natural transformations he says that isomorphism of categories is unreasonably strict for the notion of the sameness of two categories. Isomorphism would require functors,
<span class="math-container">$$
F:... | user289143 | 289,143 | <p>Two functors are said to be isomorphic if there is a natural isomorphism between them, i.e. a natural transformation <span class="math-container">$\eta: G \circ F \rightarrow 1_A$</span> such that the components <span class="math-container">$\eta_X: (G \circ F) (X) \rightarrow 1_A(X)$</span> are isomorphisms <span c... |
33,424 | <p>Suppose $f(z) = P(z)e^{Q(z)}$ where $P,Q$ are real polynomials. What is the number of non-real zeros of $f^{(k)}$ as $k$ increases?</p>
<p>We know that $f''$ has $\geq m$ zeros where $m$ depends on $Q(z)$. </p>
| John | 7,929 | <p>One could use the level sets $\{z \in H^{+}: \text{Im} Q(z) = 0 \}$ to count non-real zeros of the derivatives of $f(z)$ where $Q$ is the Newton's method function for $f$.</p>
|
859,209 | <p>Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$</p>
<p>I've tried first determining when the absolute value will be positive or negative etc, and than giving it the signing in accordance to range it is in, bur it turn... | Indrayudh Roy | 70,140 | <p>Substituting $t=|\frac {x-3}{x+1}|$ will help. Can you see it?</p>
|
859,209 | <p>Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$</p>
<p>I've tried first determining when the absolute value will be positive or negative etc, and than giving it the signing in accordance to range it is in, bur it turn... | Mark Fischler | 150,362 | <p>When $t \equiv \frac{x-3}{x+1} > 0$, the solution to $ t^2 - 7t + 10 <0$ is $ 2<t<5$. (Factor the quadratic.)</p>
<p>When $t < 0$, the solution to $ t^2 + 7t + 10 <0$ is $ -2<t<-5$. </p>
<p>On the $t < 0$ branch, we need to solve $-2 < \frac{x-3}{x+1} < 5$. We break this up into tw... |
2,370,378 | <p>How many distinct pairs of disjoint hyperplanes of size $q^{n-1}$ exist in $\mathbb{F}_q^n$?</p>
<p>Initially I had just thought to pick n points to define a hyperplane, and divide by the number of ways to pick such points, that is:</p>
<p>$\frac{q^n \choose n}{q^{n-1} \choose n}$</p>
<p>From there each hyperplan... | Angina Seng | 436,618 | <p>We call disjoint hyperplanes <em>parallel</em>. For a given hyperplane there
will be $q-1$ planes parallel to it. There are $(q^n-1)/(q-1)$
hyperplanes through the origin (why?) and so $q(q^n-1)/(q-1)$ hyperplanes overall. When counting pairs of disjoint hyperplanes I suppose
one has to decide whether one is countin... |
1,886,239 | <p>I am trying to figure out whether or not the following series is convergent: $$\sum_{k=1}^{\infty} \frac{\ln(3^k-2k)}{3k+k^2}$$ </p>
<p>Now, I know from the back of the book that it is divergent, but I haven't been able to show it. I think I am supposed to compare it to some other series, but I don't know which one... | Robert Z | 299,698 | <p>Hint: use Asymptotic Comparison Test
$$0\leq\frac{\ln(3^k-2k)}{3k+k^2}\sim \frac{\ln(3^k)}{k^2}=\frac{\ln(3)}{k}.$$</p>
|
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | guest | 12,117 | <p>Use repetition and carrot/stick (e.g. weekly period-long exams, daily one question pop quizzes, in class games). </p>
<p>Question shows an unconscious assumption that clear explanation is the key criteria. ("I told them several times.") It's not. We are not computers that get fixed forever with a line of code. ... |
126,057 | <p>Hi,</p>
<p>Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$?
This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that s... | Ahmed Sulejmani | 26,516 | <p>The answer is yes and can be found for example in S.Kobayashi "The geometry of bounded domains" T.A.M.S 1959 (92) 267- 290</p>
|
3,643,917 | <p>Find the limit of
<span class="math-container">$$\lim_{x\to 0+} \frac{1}{x^2} \int_{0}^{x} t^{1+t} dt$$</span>. </p>
<p>My idea to solve it is to use L'Hospital's rule but I am not sure why I can use it and how should i do it. Many thanks to them who are willing to help. </p>
| Paramanand Singh | 72,031 | <p>Use the substitution <span class="math-container">$u=t^2$</span> to reduce the expression under limit to <span class="math-container">$$\frac{1}{2x^2}\int_{0}^{x^2}(\sqrt {u}) ^{\sqrt{u}} \, du$$</span> By Fundamental Theorem of Calculus the desired limit equals <span class="math-container">$1/2$</span> as the integ... |
610,472 | <p>What is the value of these limits;</p>
<p>$\lim_{x\rightarrow 1^{+}}\frac{\lfloor x\rfloor-1}{\lfloor x\rfloor-x}$</p>
<p>$\lim_{x\rightarrow 1^{-}}\frac{\lfloor x\rfloor-1}{\lfloor x\rfloor-x}$</p>
| Mikasa | 8,581 | <p>Hint:</p>
<p>Look at the following figures. While $x\to 1^+$ then in a very small neighborhood of $x=1$ from the right we have: $$x\to1^+\longrightarrow x\in [1,1+\epsilon)\to \lfloor x\rfloor=1$$</p>
<p><img src="https://i.stack.imgur.com/PJRjQ.jpg" alt="enter image description here"></p>
<p><img src="https://i.... |
3,624,953 | <p>Let <span class="math-container">$(X_t)_{t \in [0, T]}$</span> be a continuous stochastic process with paths which are a.s. continuous, the underlying space of which is irrelevant but is well defined. Let <span class="math-container">$a$</span> be a constant
Define two stopping times
<span class="math-container">$$\... | Kavi Rama Murthy | 142,385 | <p>You need some assumption on <span class="math-container">$X_0$</span> for this. Assume that <span class="math-container">$X_0 <a$</span>. Then you can apply the following elementary fact:</p>
<p>Let <span class="math-container">$f:[0,\infty) \to \mathbb R$</span> be a continuous function with <span class="math-c... |
102,932 | <p>This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) </p>
<p>I think I can figur... | Ian Agol | 1,345 | <p>If the group <span class="math-container">$G$</span> does not have <a href="http://en.wikipedia.org/wiki/Property_FA" rel="nofollow noreferrer">property FA</a>, then a necessary and sufficient
condition is that the group embeds in <span class="math-container">$\operatorname{GL}_2(\mathcal{O}_K)$</span>, for some num... |
102,932 | <p>This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) </p>
<p>I think I can figur... | Misha | 21,684 | <p>First, instead of subgroups of $GL(2,C)$ it suffices to consider subgroups of $SL(2, C)$. I will also restrict to subgroups which are not virtually solvable (since these should be easy to classify). Lastly, I will consider characterization of subgroups of $SL(2, C)$ "up to abstract commensuration" since the criterio... |
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| Sabino Di Trani | 82,009 | <p>Let's try with substitution $x=\frac{t}{\log a}$</p>
<p>Using $a=e^{\log a}$ you obtain $$\lim_{t\to \infty} \left(\frac{a^{\frac{t}{\log a}}-1}{t}\right) \log a= \lim_{t\to \infty} \left(\frac{e^t-1}{t}\right) \log a= \log a$$</p>
|
257,027 | <p><strong>Question:</strong> Consider a distribution $D$, and $n$ i.i.d. random variables $X_i$, all distributed according to $D$. Let $p^D_2:=\Pr[X_1=X_2]$. What is a lower bound for $p^D_n:=\Pr[\exists i\neq j. X_i=X_j]$ (as a function of $p^D_2$)?</p>
<p><strong>Conjecture:</strong> $p^D_n \geq 1-\bigl(1-p^D_2\big... | zhoraster | 8,146 | <p>Let $X_i$ take values $x_1,x_2,\dots$ with probabilities $p_1,p_2,\dots$</p>
<p>Define the events
$$
A_i = \{\exists j\neq i: X_i = X_j\}
$$
By the Chung-Erdős inequality,
$$
p^D_n = P\left(\bigcup_{i=1}^n A_i\right) \ge \frac{\big(\sum_{i=1}^n P(A_i)\big)^2}{\sum_{i=1}^n P(A_i) + \sum_{i\neq j}P(A_i\cap A_j)}\\ ... |
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | Zarrax | 3,035 | <p>You can always rotate the sphere so that points A and B are both on the equator. The idea then is you reduce your distance from point B the fastest if you head in the direction of point B, and that direction is along the equator. </p>
|
1,818,557 | <p>I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?</p>
<p>Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot fin... | Martin Kochanski | 340,970 | <p>Use a ball. Note that the less curvy a line is, the straighter it is (and therefore shorter). </p>
<p>Then note that cutting a slice through the <strong>middle</strong> of the ball gets you the straightest line available. Which is a good definition of a great circle.</p>
<p><em>(And as other answerers have pointed... |
1,133,694 | <p>Suppose $a_{1},a_{2},a_{3}...a_{n}$ is a complex sequence satisfying $\bigl\lvert\left(\sum_{{k=1}}^{n}a_{k}b_{k}\right)\bigr\rvert \leq1$ for all $b_1,b_2,...,b_n$ such that $\left(\sum_{{k=1}}^{n}\mid b_{k}\mid^{2}\right)\leq1$. Show that $\left(\sum_{{k=1}}^{n}\mid a_{k}\mid^{2}\right)\leq1$</p>
<p>I'm consideri... | WimC | 25,313 | <p>If $a=0$ then the inequality holds. Now suppose $\lVert a \rVert=\left(\sum|a_j|^2\right)^{1/2} > 0$. Let $b_j = \overline{a_j}/\lVert a \rVert$. Then $\sum |b_j|^2=1$ and $\lVert a \rVert = \sum a_j b_j \leq 1 $.</p>
|
1,133,694 | <p>Suppose $a_{1},a_{2},a_{3}...a_{n}$ is a complex sequence satisfying $\bigl\lvert\left(\sum_{{k=1}}^{n}a_{k}b_{k}\right)\bigr\rvert \leq1$ for all $b_1,b_2,...,b_n$ such that $\left(\sum_{{k=1}}^{n}\mid b_{k}\mid^{2}\right)\leq1$. Show that $\left(\sum_{{k=1}}^{n}\mid a_{k}\mid^{2}\right)\leq1$</p>
<p>I'm consideri... | Zoe | 212,300 | <p>Suppose $\textbf{a,b}$ are complex vectors such that $\textbf{a}=(a_1,a_2,...,a_n) and \textbf{b}=(b_1,b_2,...,b_n)$.</p>
<p>Then $|\textbf{a}|^2=\sum_{k=1}^{n}|a_k|^2$ and $|\textbf{b}|^2=\sum_{k=1}^{n}|b_k|^2$.</p>
<p>Start proof by supposing $|\textbf{a}|^2 >1$.</p>
<p>Let $\textbf{b}=\frac{\textbf{a}}{|\t... |
1,950,466 | <p>Let $x_1, x_2, ...,x_n \in \mathbb{H}$, $\mathbb{H}$ is the Hilbert space. $x_j \neq 0$ for every $j$. If $x_i \perp x_j$ for $i \neq j$ then show that $x_j$ 's are linearly independent. </p>
<p>I think I need to use the fact that $<\alpha x + \beta y , z> = \alpha<x,z> + \beta<y,z>$ but how?</p... | DonAntonio | 31,254 | <p>Suppose $\;\sum\limits_{i=1}^na_ix_i=0\;$ , then for all $\;1\le k\le n\;$ :</p>
<p>$$\left\langle 0,x_k\right\rangle=\left\langle \sum\limits_{i=1}^na_ix_i\,,\,\,x_k\right\rangle=\sum\limits_{i=1}^na_i\langle x_i,\,x_k\rangle=a_k\langle x_k,\,x_k\rangle$$</p>
<p>End the proof now.</p>
|
4,651,798 | <p>This question comes from the province-stage olympiad in my country in order to qualify for the national stage:</p>
<blockquote>
<p>Given the set <span class="math-container">$S=\{1,2,3,4\}$</span>.The number of non-empty subsets <span class="math-container">$A_1,A_2, ..., A_6$</span> that fulfil these three criteria... | Marc van Leeuwen | 18,880 | <p>It seems simplest to me to put <span class="math-container">$B_2=A_2\cup A_1$</span> and <span class="math-container">$B_i=A_i$</span> for <span class="math-container">$i\in\{1,3,4,5,6\}$</span>; we can recover the <span class="math-container">$A_i$</span> from the <span class="math-container">$B_i$</span> by <span ... |
4,202,654 | <p>This question regards the proof of proposition 8 in chapter 8.1 of Bosch's 'Algebraic Geometry and Commutative Algebra'.</p>
<p>Let <span class="math-container">$A, B$</span> be <span class="math-container">$R$</span>-algebras, <span class="math-container">$f_0: A \to B$</span> a morphism of <span class="math-contai... | Mark Saving | 798,694 | <p>I think an easier proof is to note that <span class="math-container">$f_1 - f_0 \in \mathfrak{I}$</span>. Then <span class="math-container">$0 = (f_1(1) - f_0(1))^2 = (f_1(1) - 1)^2 = f_1(1)^2 - 2 f_1(1) + 1$</span>, so <span class="math-container">$1 = 2 f_1(1) - f_1(1)^2$</span>. Applying the multiplicative proper... |
4,202,654 | <p>This question regards the proof of proposition 8 in chapter 8.1 of Bosch's 'Algebraic Geometry and Commutative Algebra'.</p>
<p>Let <span class="math-container">$A, B$</span> be <span class="math-container">$R$</span>-algebras, <span class="math-container">$f_0: A \to B$</span> a morphism of <span class="math-contai... | Milten | 620,957 | <p>You have <span class="math-container">$f_1(1) = f_0(1)+x = 1+x$</span> for some <span class="math-container">$x\in\mathfrak I$</span>. Then
<span class="math-container">$$
f_1(1)(1-x) = 1-x^2 = 1,
$$</span>
since <span class="math-container">$\mathfrak I^2=0$</span>, so <span class="math-container">$f_1(1)$</span> i... |
789,802 | <p>How do I solve $\displaystyle \lim_{x \to 0} \sqrt{x^2 + x^3} \sin \frac{\pi}{x}$ using squeeze Theorem?</p>
<p>My book only teaches me the simplest use of the Theorem. I have no idea what should I do with a function as complex as this...</p>
<p>I know I have to start with:</p>
<p>$$ -1 \le \sin \frac{\pi}{x} \le... | Ivo Terek | 118,056 | <p>You have $$-1 \leq \sin \frac{\pi}{x} \leq 1$$</p>
<p>Multiply it by what is left:</p>
<p>$$-\sqrt{x^2 + x^3} \leq \sqrt{x^2 + x^3} \sin \frac{\pi}{x} \leq \sqrt{x^2 + x^3}$$</p>
<p>What happens when $x \to 0$? The limit we want will be <em>squeezed</em> between which values? </p>
|
2,721,372 | <p>A sequence is defined by $a_1=2$ and $a_n=3a_{n-1}+1 $ .Find the sum $a_1+a_2+\cdots+a_n$</p>
<p>how to find sum $a_1=2,a_2=7,\ldots$</p>
<p>Also i found the value of $a_n=\frac{5}{6}\cdot3^n-\frac{1}{2}$</p>
| Jordan | 545,687 | <p>In general, the predicate logic statement that $\forall x(x \in A \implies x \in B)$ is written as $A \subseteq B$.</p>
<p>The <em>empty set</em> $\emptyset$ is the set that contains no elements. Therefore, the empty set is a subset of any set, that is, $\emptyset \subseteq X$ for all $X$. This is because the state... |
1,348,587 | <p>Let $f$ be a real-valued function (a function with target space the set of reals). Let $P(x, M)$ stand for $|f(x)| \leq M $, let $N$ be the set of positive real numbers, and let $\mathbb{R}$ be the set of real numbers.</p>
<p>a) Which of the following statements is an accurate translation of "f is bounded"?</p>
<p... | Mauro ALLEGRANZA | 108,274 | <p>The difference between : $\exists x \forall y$ and $\forall y \exists x$ is clearly shown by this example regarding <em>natural numbers</em> :</p>
<blockquote>
<p>$\forall n \ \exists m \ (n < m)$</p>
</blockquote>
<p>is clearly true in $\mathbb N$,</p>
<p>while :</p>
<blockquote>
<p>$\exists m \ \forall ... |
1,715,032 | <p>When given propositions to prove such as the following question:
prove that $|z+i| = |z-i|$ if $z \in \mathbb{R}$.</p>
<p>Would I have to prove this proposition without substituting $z$ for a complex number?</p>
| Community | -1 | <p>You should certainly NOT substitute a complex number for $z$ (not the other way around, anyway!) if you are asked to prove the identity for REAL valued $z$.</p>
<p>So - rephrasing - yes, you must prove that proposition without substituting some random real numbers for $z$. That will only prove the proposition for t... |
2,783,129 | <p>How many different value of x from 0° to 180° for the equation $(2\sin x-1)(\cos x+1) = 0$?</p>
<p>The solution shows that one of these is true:</p>
<p>$\sin x = \frac12$ and thus $x = 30^\circ$ or $120^\circ$ </p>
<p>$\cos x = -1$ and thus $x = 180^\circ$</p>
<p><strong>Question:</strong> Inserting the $\arcsin... | user | 505,767 | <p>As you noted $x=120°$ is not a solution indeed</p>
<ul>
<li>$(2\sin 120°-1)(\cos 120°+1)=\left(2\frac{\sqrt3}2-1\right)\left(-\frac12+1\right)\neq 0$</li>
</ul>
<p>but</p>
<ul>
<li>$2\sin x-1=0 \implies x=\frac16 \pi + 2k\pi,\,x=\frac56 \pi + 2k\pi$</li>
<li>$\cos x+1 = 0 \implies x=\pi + 2k\pi=(2k+1)\pi$</li>
</... |
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