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 |
|---|---|---|---|---|
3,527,004 | <p>As stated in the title, I want <span class="math-container">$f(x)=\frac{1}{x^2}$</span> to be expanded as a series with powers of <span class="math-container">$(x+2)$</span>. </p>
<p>Let <span class="math-container">$u=x+2$</span>. Then <span class="math-container">$f(x)=\frac{1}{x^2}=\frac{1}{(u-2)^2}$</span></p>
... | vonbrand | 43,946 | <p>Binomial theorem:</p>
<p><span class="math-container">$\begin{align*}
\frac{1}{x^2}
&= (2 + (x - 2))^{-2} \\
&= \frac{1}{4}
\cdot \left(
1 + \frac{1}{2}(x - 2)
\right)^{-2} \\
&= \frac{1}{4}
\cdot \sum_{k \ge ... |
1,694,159 | <p>I am prepping for my mid semester exam, and came across with the following question:</p>
<blockquote>
<p>Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that $k = 0, 1,...n$ and $n$ can be any natural number.</p>
</blockquote>
<p>So what I have done is to ... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\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{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
281,294 | <p>Let $G$ be a finite abelian group.</p>
<p>Is there a field $K$, and an elliptic curve $E$ over $K$ such that $E(K)_{tor} \cong G$?</p>
| David Lampert | 59,248 | <p>If $n \geq 1$ and $2|mn$ then the Tate curve $E = L^*/2^{\mathbb Z}$ with $L={\mathbb Q}_2(\zeta_{mn}, 2^{1/n})$ has $E(L)_{tor} = {\mathbb Z}/n{\mathbb Z} \times {\mathbb Z}/mn{\mathbb Z}$. A 2-adically close elliptic curve over a suitable (i.e. containing the curve's corresponding torsion) number field $K \subse... |
850,852 | <p>This one comes from Gilbert Strang's Linear Algebra. Pick any numbers $x+y+z = 0$. Find an angle between $\mathbf v=(x,y,z)$ and $\mathbf w=(z,x,y)$. </p>
<p>Explain why $$\dfrac{\bf v\cdot w}{\bf \Vert v\Vert \cdot\Vert w\Vert}$$ is always $-0.5$. </p>
| Christopher K | 101,768 | <p>Let $(***)$ be the equation, then $$(***) = \frac{xz+xy+yz}{x^2+y^2+z^2} = \frac{x(-x)+yz}{x^2+y^2+z^2}
\\ = \frac{-x^2 + 1/2\cdot[(y+z)^2-y^2-z^2]}{x^2+y^2+z^2} \\
= \frac{-x^2 + 1/2\cdot[x^2 - y^2 - z^2]}{x^2+y^2+z^2} \\
= \frac{(-1/2)\cdot (x^2+y^2+z^2)}{x^2+y^2+z^2} \\ = -1/2,$$ as desired.</p>
|
1,885,177 | <p>What is the meaning of $x$ raised to any non-positive value? We know that $x^{-a} = \dfrac{1}{x^a}$ and $x^0 = 1$, but where does that come from? What is the proof? Why is this true? What about $x$ raised to a fraction, like say $\frac{1}{3}$? <em>How do you multiply $x$ $\frac{1}{3}$ times?</em></p>
| Simply Beautiful Art | 272,831 | <p>We know that $x^{a+b}=x^ax^b$ since an $(a+b)$ amount of $x$'s is the same as an $a$ amount of $x$'s times a $b$ amount of $x$'s.</p>
<p>From this, we have</p>
<p>$$x^0=x^{a-a}=x^ax^{-a}$$</p>
<p>Thus, we have</p>
<p>$$x^{-a}=\frac{x^0}{x^a}$$</p>
<p>and once you show $x^0=1$, your done.</p>
<p>As for fraction... |
1,885,177 | <p>What is the meaning of $x$ raised to any non-positive value? We know that $x^{-a} = \dfrac{1}{x^a}$ and $x^0 = 1$, but where does that come from? What is the proof? Why is this true? What about $x$ raised to a fraction, like say $\frac{1}{3}$? <em>How do you multiply $x$ $\frac{1}{3}$ times?</em></p>
| Arthur | 15,500 | <p>From that definition, it doesn't make sense. It's what we mathematicians call a generalisation. The original definition of powers only allow natural numbers to be exponents.</p>
<p>However, one can ask "Is it possible to come up with a definition of $x^{-a}$ that plays well along with the old definition?" By which ... |
1,514,094 | <p>Given three points on the $xy$ plane on $O(0,0),A(1,0)$ and $B(-1,0)$.Point $P$ is moving on the plane satisfying the condition $(\vec{PA}.\vec{PB})+3(\vec{OA}.\vec{OB})=0$<br>
If the maximum and minimum values of $|\vec{PA}||\vec{PB}|$ are $M$ and $m$ respectively then prove that the value of $M^2+m^2=34$</p>
<hr>... | cr001 | 254,175 | <p>The answer is correct and you are wong.</p>
<p>The mistake in your solution happens when you apply the Cauchy Inequality where you missed to square the right side.</p>
<p>$(x_1^2+x^2_2+x_3^2+x_4^2+x_5^2)(1+1+1+1+1)\ge(x_1+x_2+x_3+x_4+x_5)^2=400$</p>
<p>So $x_1^2+x^2_2+x_3^2+x_4^2+x_5^2\ge\frac{400}{5}=80$</p>
<p... |
1,514,094 | <p>Given three points on the $xy$ plane on $O(0,0),A(1,0)$ and $B(-1,0)$.Point $P$ is moving on the plane satisfying the condition $(\vec{PA}.\vec{PB})+3(\vec{OA}.\vec{OB})=0$<br>
If the maximum and minimum values of $|\vec{PA}||\vec{PB}|$ are $M$ and $m$ respectively then prove that the value of $M^2+m^2=34$</p>
<hr>... | DuFong | 193,997 | <p>The condition for your answer is when $x_1=x_2=...=x_5=4$,so we can find that they are roots of $$(x-4)^5=0$$</p>
<p>Expand this equation of left side we have the coefficient of $x^4$ is $$C(5,4)*(-4)^1=-20$$
and coefficient of $x^3$ is $$C(5,3)*(-4)^2=160$$</p>
|
1,400,399 | <p>Here is an indefinite integral that is similar to an integral I wanna propose for a contest. Apart from
using CAS, do you see any very easy way of calculating it?</p>
<p>$$\int \frac{1+2x +3 x^2}{\left(2+x+x^2+x^3\right) \sqrt{1+\sqrt{2+x+x^2+x^3}}} \, dx$$</p>
<p><strong>EDIT:</strong> It's a part from the gener... | robjohn | 13,854 | <p>Let $\sinh^4(t)=x^3+x^2+x+2$. Then $\cosh(t)=\sqrt{1+\sqrt{x^3+x^2+x+2}}$ and
$$
\begin{align}
&\int\frac{3x^2+2x+1}{\left(x^3+x^2+x+2\right)\sqrt{1+\sqrt{x^3+x^2+x+2}}} \,\mathrm{d}x\\
&=\int\frac{\mathrm{d}\sinh^4(t)}{\sinh^4(t)\cosh(t)}\\
&=4\int\frac{\mathrm{d}t}{\sinh(t)}\\
&=4\int\frac{\mathrm{... |
540,135 | <p>$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?</p>
<p>I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... </p>
| lab bhattacharjee | 33,337 | <p>Let the highest power of prime $p$ in $a,b,c$ be $A,B,C$ respectively.</p>
<p>The highest power of prime $p$ in lcm$(a,b,c)=$max$(A,B,C)$</p>
<p>The highest power of prime $p$ in lcm$(a,b)=$max$(A,B)$</p>
<p>The highest power of prime $p$ in lcm$($lcm$(a,b),c)=$max$($max$(A,B),C)$</p>
<p>Can you see max$($max$(A... |
540,135 | <p>$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?</p>
<p>I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... </p>
| tchappy ha | 384,082 | <blockquote>
<p>Lemma:<br />
Let <span class="math-container">$l:=\operatorname{lcm}(a,b,\dots,z)$</span>.<br />
Let <span class="math-container">$m$</span> be a common multiple of <span class="math-container">$a,b,\dots z$</span>.<br />
Then, <span class="math-container">$m$</span> is a multiple of <span class="math-c... |
2,986,647 | <p>If I know coordinates of point <span class="math-container">$A$</span>, coordinates of circle center <span class="math-container">$B$</span> and <span class="math-container">$r$</span> is the radius of the circle, is it possible to calculate the angle of the lines that are passing through point A that are also tange... | Fnacool | 318,321 | <p>Here's a different approach. </p>
<p>1) <span class="math-container">$$E[e^{-s|B_t|^2}] = E[e^{-ts N(0,1)^2}]^3.$$</span> </p>
<p>2) <span class="math-container">\begin{align*} E[e^{-u N(0,1)^2}] & = \frac{1}{\sqrt{2\pi}}\int e^{-x^2 ( u + 1/2) }dx\\
& = (2u+1)^{-1/2}.
\end{align*}</span> </p>
<p>3) <spa... |
3,127,459 | <p>If
α
,
β
,
γ
</p>
<p>are the roots of equation <span class="math-container">$x^3 -x -1 =0$</span>
then</p>
<p><span class="math-container">$$ \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma} $$</span></p>
<p>My attempt is in the attachme... | David Quinn | 187,299 | <p>hint...write <span class="math-container">$$y=\frac{1+x}{1-x}\implies x=\frac{y-1}{y+1}$$</span> and substitute into the polynomial. Simplify the polynomial in <span class="math-container">$y$</span> and find the sum of the roots.</p>
<p>Alternatively,<span class="math-container">$$\Sigma\frac{1+\alpha}{1-\alpha}=-... |
3,127,459 | <p>If
α
,
β
,
γ
</p>
<p>are the roots of equation <span class="math-container">$x^3 -x -1 =0$</span>
then</p>
<p><span class="math-container">$$ \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma} $$</span></p>
<p>My attempt is in the attachme... | DXT | 372,201 | <p><span class="math-container">$$\sum \frac{1+\alpha}{1-\alpha}=2\sum\frac{1}{1-\alpha}-3\cdots (1)$$</span></p>
<p>Where <span class="math-container">$$\sum\frac{1+\alpha}{1-\alpha}=\frac{1+\alpha}{1-\alpha}+\frac{1+\beta}{1-\beta}+\frac{1-\gamma}{1-\gamma}.$$</span></p>
<p>Now adding <span class="math-container">$... |
3,264,333 | <p>I am working on my scholarship exam practice and not sure how to begin. Please assume math knowledge at high school or pre-university level.</p>
<blockquote>
<p>Let <span class="math-container">$a$</span> be a real constant. If the constant term of <span class="math-container">$(x^3 + \frac{a}{x^2})^5$</span> is ... | paulinho | 474,578 | <p>From the binomial theorem, we know that the sum of the powers of <span class="math-container">$x^3$</span> and <span class="math-container">$a/x^2$</span> must add to <span class="math-container">$5$</span>. So if our term in the expansion is <span class="math-container">$k (x^3)^p (\frac{a}{x^2})^q$</span>, <span c... |
2,962,193 | <p><strong>Q</strong>:If <span class="math-container">$2\cos p=x+\frac{1}{x}$</span> and <span class="math-container">$2\cos q=y+\frac{1}{y}$</span> then show that <span class="math-container">$2\cos(mp-nq)$</span> is one of the values of <span class="math-container">$\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$</sp... | David G. Stork | 210,401 | <p>If <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$f(x,y)$</span> are integers or integer-valued, then there are more inputs than available outputs. By the pigeonhole principle, no such function exists.</p>
<p>If <span class="math-container">$f$</s... |
1,829,342 | <p>So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone point me to a formula with proof for these two sums? My searches thus far have only turned up those first two sums wi... | Hao S | 202,187 | <p>So for n even you can use $ {n \choose 2i} = {n \choose n-2i}$ to rewrite the sum as
$\sum_i i {n \choose n-2i}$ then add this sum to the original sum to get $\sum_i n{n \choose n-2i}$</p>
|
285,548 | <p>I asked the following question on math.SE (<a href="https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d">https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d</a>) just over t... | Bombyx mori | 18,850 | <p><a href="http://www.maths.ed.ac.uk/~aar/papers/melrose.pdf" rel="nofollow noreferrer">The book</a> by Melrose was almost exactly written to address issues like the ones you mentioned above, where the very first example is the one in the question. The construction fo the parametrix is the focus point of the book. How... |
2,560,556 | <p>Let $X,Y,Z$ be topological spaces.
Let $p:X\rightarrow Y$ be a continuous surjection. Let $f:Y\rightarrow Z$ be continuous if and only if $f\circ p:X\rightarrow Z$ is continuous.</p>
<p>I want to prove that this makes $p$ a quotient map. </p>
<p>My thoughts:</p>
<p>Since $p$ is a continuous surjection, all I need... | Henno Brandsma | 4,280 | <p>So $p:X \to Y$ obeys the property that </p>
<blockquote>
<p>for all functions $g: Y \to Z$, $g$ is continuous iff $g \circ p$ is continuous.</p>
</blockquote>
<p>Then suppose that $U$ is a subset of $Y$ that satisfies $p^{-1}[U]$ is open in $X$. Then define $Z = \{0,1\}$ with the topology $\{\{0\}, \emptyset, Z\... |
3,432,911 | <p>My argument is as follows:</p>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$I$</span> an ideal of <span class="math-container">$R$</span>.
If <span class="math-container">$(R/I)^n\cong (R/I)^m$</span> as <span class="math-container">$R$</span>-module... | José Carlos Santos | 446,262 | <p>Note that, for each <span class="math-container">$z\in\mathbb C$</span>,<span class="math-container">\begin{align}\lvert z-1\rvert=1&\iff\lvert z-1\rvert^2=1\\&\iff\lvert z\rvert^2-2\operatorname{Re}(z)+1=1\\&\iff\lvert z\rvert^2=2\operatorname{Re}(z).\end{align}</span>So, if <span class="math-container"... |
3,432,911 | <p>My argument is as follows:</p>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$I$</span> an ideal of <span class="math-container">$R$</span>.
If <span class="math-container">$(R/I)^n\cong (R/I)^m$</span> as <span class="math-container">$R$</span>-module... | Quanto | 686,284 | <p>Let <span class="math-container">$z-1 = e^{ia}$</span>, then</p>
<p><span class="math-container">$$\frac 1z = \frac 1{1+e^{ia}}=\frac {1+e^{-ia}}{(1+e^{ia})(1+e^{-ia})}=\frac {1+\cos a - i\sin a}{2+2\cos a}=\frac 12 -\frac12\tan\frac a2 i $$</span></p>
<p>Thus, <span class="math-container">$\frac 1z$</span> is of ... |
3,424,189 | <p>I'm trying to calculate the integral <span class="math-container">$$\int_0^1 \frac{\sin\Big(a \cdot \ln(x)\Big)\cdot \sin \Big(b \cdot \ln(x)\Big)}{\ln(x)} dx, $$</span>
but am stuck. I tried using Simpsons' rules and got here:
<span class="math-container">$$\int_0^1 \frac{\cos\Big((a+b) \cdot \ln(x)\Big) - \cos \Bi... | Claude Leibovici | 82,404 | <p>I do not think that it would be very pleasant.</p>
<p>After your simplification, you face two integrals looking like
<span class="math-container">$$I=\int \frac {\cos(k \log(x))} {\log(x)} \,dx$$</span> First, let <span class="math-container">$x=e^t$</span> to make
<span class="math-container">$$I=\int \frac{e^t \c... |
673,334 | <p>I used below pseudocode to generate a discrete normal distribution over 101 points.</p>
<pre><code>mean = 0;
stddev = 1;
lowerLimit = mean - 4*stddev;
upperLimit = mean + 4*stddev;
interval = (upperLimit-lowerLimit)/101;
for ( x = lowerLimit + 0.5*interval ; x < upperLimit; x = x + interval) { ... | StasK | 97,144 | <p>The height of your "density" <code>y</code> should account for the width of the interval that you have discretized your distribution to. In other words, you need to assign the mass $\int_{x_0}^{x_0+h} \phi(x) \, {\rm d}x = \Phi(x_0+h) - \Phi(x_0) \approx \phi(x_0) h$ to the point $x_0$, where $\phi(x)$ is the standa... |
8,568 | <p>I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. </p>
<p>I don't want to start the class out with "getting down to business" beca... | JRN | 77 | <p>The article titled "Math Teachers’ Circles: Partnerships between Mathematicians and Teachers" by B. Donaldson, M. Nakamaye, K. Umland, and D. White in the December 2014 <em>Notices of the AMS</em> (pp. 1335-1341) (<a href="http://www.ams.org/notices/201411/rnoti-p1335.pdf" rel="nofollow noreferrer">pdf ver... |
8,568 | <p>I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. </p>
<p>I don't want to start the class out with "getting down to business" beca... | kcrisman | 1,608 | <p>There are many, many examples of this in the resources at <a href="https://www.artofmathematics.org/" rel="nofollow">The Art of Discovering Mathematics</a>, including things like the game of Hex.</p>
|
2,543,834 | <p>Ok, so in my differential equations class we've been doing problems which more or less amount to solving equations of the form:</p>
<p><span class="math-container">$$\frac{dY}{dt} = AY$$</span></p>
<p>Where <span class="math-container">$A$</span> is just some <span class="math-container">$2\times2$</span> linear tra... | Math Lover | 348,257 | <p>Observe that $A = S \Lambda S^{-1}$, where $S = \begin{bmatrix}V_1 & V_2\end{bmatrix}$, $\Lambda = \begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix}$, and $S^{-1}=S^T$. Consequently,
$$\frac{dY(t)}{dt}=AY(t) = S\Lambda S^{-1} Y(t). \tag{1}$$
Let $S^{-1}Y(t)=Z(t)$. Consequently, $(1)$ becomes
$$\... |
2,543,834 | <p>Ok, so in my differential equations class we've been doing problems which more or less amount to solving equations of the form:</p>
<p><span class="math-container">$$\frac{dY}{dt} = AY$$</span></p>
<p>Where <span class="math-container">$A$</span> is just some <span class="math-container">$2\times2$</span> linear tra... | Disintegrating By Parts | 112,478 | <p>Consider the vector ODE
$$
\frac{d Y}{dt} = AY,\;\; Y(0)=Y_0,
$$
where $Y$ is an $N$-vector function of $y$, where $Y_0$ is a constant $N$-vector, and where $A$ is a constant $N\times N$ matrix. This has a unique solution
$$
Y(t) = e^{tA}Y_0\; \mbox{where}\; e^{tA}=\sum_{n=0}^{\infty}\frac{1}{n!}t^... |
9,930 | <p>One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?</p>
<p>To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are ... | David E Speyer | 297 | <p>I'm not sure this is what you are looking for, but if you look at Gelfand and Manin's <em>Methods of Homological Algebra</em>, the end of Section II.5, they make the following definition:</p>
<p>Let $Y$ be an object of an abelian category. An <em>element</em> of $Y$ is a pair $(X,h)$, with $h: X \to Y$, modulo the ... |
2,885,097 | <p>I'm having difficulty calculating the Taylor series of $\frac{1}{1-x}$ about $a=3$, and was wondering if anyone on here could help me out.</p>
<p>Here's what I've tried so far:</p>
<p><strong>Attempt 1 -</strong> Take $y=x-3$ and rearrange as $x=y+3$.</p>
<p>Then do $\frac{1}{1-x}=\frac{1}{1-(y+3)}=\frac{1}{-y-2}... | José Carlos Santos | 446,262 | <p>Your first attempt is wrong, because after you do the substitution $y=x-3$, you should aim at a power series about $0$, not about $-1$.</p>
<p>Your second attempt is corect (but I would put $(-1)^n$ in the numerator and $2^n$ in the denominator).</p>
|
2,885,097 | <p>I'm having difficulty calculating the Taylor series of $\frac{1}{1-x}$ about $a=3$, and was wondering if anyone on here could help me out.</p>
<p>Here's what I've tried so far:</p>
<p><strong>Attempt 1 -</strong> Take $y=x-3$ and rearrange as $x=y+3$.</p>
<p>Then do $\frac{1}{1-x}=\frac{1}{1-(y+3)}=\frac{1}{-y-2}... | Jean-Claude Arbaut | 43,608 | <p>As already said above, your first attempt is incorrect. But it's easy to do it correctly: for $|y|<2$,</p>
<p>$$\frac{1}{1-(y+3)}=-\frac{1}{2+y}=-\frac12\frac{1}{1+\frac{y}2}=-\frac12\sum_{n=0}^\infty (-1)^n\frac{y^n}{2^n}$$</p>
|
3,687,043 | <p>I have a polynomial <span class="math-container">$f(z)=z^4+z^3-2z^2+2z+4$</span>, and I want to find the number of roots in the first quadrant. I'm trying to use the argument principle (or Rouche), and I could try to make my contour the quarter circle, but I've having trouble because I can't justify that there are n... | Sarvesh Ravichandran Iyer | 316,409 | <p>This one is a real weird one. </p>
<p>That is because I am tempted to factor this polynomial.</p>
<p>Let us try the rational root theorem first. If there is a rational root <span class="math-container">$\frac pq$</span> then <span class="math-container">$p$</span> divides <span class="math-container">$4$</span> an... |
3,687,043 | <p>I have a polynomial <span class="math-container">$f(z)=z^4+z^3-2z^2+2z+4$</span>, and I want to find the number of roots in the first quadrant. I'm trying to use the argument principle (or Rouche), and I could try to make my contour the quarter circle, but I've having trouble because I can't justify that there are n... | Oscar Lanzi | 248,217 | <p>You can use the Rouche method if you prove that there are no roots on the <em>positive</em> real axis, which is the part of the real axis that's on your path.</p>
<p>When <span class="math-container">$x>\sqrt2$</span> (I'm using <span class="math-container">$x$</span> here to emphasize we're dealing with a real ... |
237,446 | <p>I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$
what gives $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \geqslant \lim_{n\rightarrow\inft... | user 1591719 | 32,016 | <p>Since it's so hard let's solve it in one line</p>
<p>$$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )=-\lim_{n\rightarrow\infty}\left(\frac{\sqrt[n]{\ln(n)}-1}{\displaystyle\frac{1}{n}\ln(\ln (n)) }\cdot \ln(\ln (n))\right)=-(1\cdot \infty)=-\infty.$$</p>
<p>Chris.</p>
|
181,940 | <p>I've been unable to find an answer to the following question in the literature
on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$
where $\kappa$ is inaccessible. The basic open sets are the $U_f$ where
$f\in\kappa^{<\kappa}$. A perfect set is a nonempty closed set with no
isolated po... | Joel David Hamkins | 1,946 | <p>It is consistent to have a perfect set of size
$\kappa$, or of intermediate size between $\kappa$ and $2^\kappa$. </p>
<p>To see this, suppose that $\kappa$ is an inaccessible cardinal in $V$, and let
$T=2^{<\kappa}$ be the tree of ${<}\kappa$ binary sequences, and
let $X=[T]={}^\kappa 2$ be the set of branch... |
417,064 | <p>Let T be a totally ordered set that is <strong>finite</strong>. Does it follow that minimum and maximum of T exist?
Since T is finite, I believe there exists a minimal of T. From that it maybe able to be shown that the minimal is the minimum but not quite sure whether it is the right approach. </p>
| Community | -1 | <p>If a totally ordered set is finite <strong>and nonempty</strong>, then it follows that it has a maximum and a minimum.</p>
|
4,486,131 | <p><span class="math-container">$F=Q(\sqrt{2i})$</span>,then
Which one of the following is not true (Duet-2017 Q.26)</p>
<p>1.<span class="math-container">$\sqrt{2}\in F$</span></p>
<p>2.<span class="math-container">$i \in F$</span></p>
<p>3.<span class="math-container">$x^8-16=0$</span> has a solution in <span class="... | Guillermo García Sáez | 696,501 | <ol>
<li><p>Is false, in case it's true we would have <span class="math-container">$F(\sqrt2)=F$</span>, but
<span class="math-container">$$F(\sqrt2)=\mathbb{Q(\sqrt{2i},\sqrt 2)}=\mathbb{Q(\sqrt i,\sqrt 2)}\not =\mathbb{Q(\sqrt{2i})}=F$$</span></p>
</li>
<li><p>Similar argument</p>
</li>
<li><p>Is true
<span class="ma... |
3,998,018 | <p>I was preparing for a calculus exam and I came across the Wikipedia <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities" rel="nofollow noreferrer">article</a> for all the trigonometric identities. There I came across some terms that I had never seen before. They were:
<span class="math-container"... | Community | -1 | <p>The versine function is well documented in this Wikipedia article: <a href="https://en.wikipedia.org/wiki/Versine" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Versine</a></p>
<blockquote>
<p>The versine or versed sine is a trigonometric function found in some of the earliest (Vedic Aryabhatia I) trigonom... |
3,998,018 | <p>I was preparing for a calculus exam and I came across the Wikipedia <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities" rel="nofollow noreferrer">article</a> for all the trigonometric identities. There I came across some terms that I had never seen before. They were:
<span class="math-container"... | open problem | 876,065 | <p>As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.</p>
<p>These days we walk ar... |
3,354,684 | <p>I was trying to prove following inequality:</p>
<p><span class="math-container">$$|\sin n\theta| \leq n\sin \theta \
\text{for all n=1,2,3... and } \
0<\theta<π $$</span></p>
<p>I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?</p>
| albert chan | 696,342 | <p>Let <span class="math-container">$z = \cos(θ) + i \sin(θ) $</span></p>
<p><span class="math-container">$\displaystyle
{\sin(n\,θ)\over \sin(θ)}
= {\Im(z^n) \over \Im(z)}
= {z^n - 1/z^n \over z - 1/z}
= z^{n-1} + z^{n-3}+\cdots+{1\over z^{n-3}} + {1\over z^{n-1}}$</span></p>
<p>RHS can be group in pairs (except po... |
121,897 | <p>I want to check if a user input the function with all the specified variables or not. For that I choose the replace variables with some values and check for if the result is a number or not via a doloop. I am thinking there might be more elegant way of doing it such as <a href="http://reference.wolfram.com/language... | J. M.'s persistent exhaustion | 50 | <p>Using an <em>undocumented</em> function:</p>
<pre><code>Reduce`FreeVariables[u] === Sort[vas]
</code></pre>
|
381,036 | <p>I must show that $f(x)=p{\sqrt{x}}$ , $p>0$ is continuous on the interval [0,1). </p>
<p>I'm not sure how I show that a function is continuous on an interval, as opposed to at a particular point. </p>
| moray95 | 68,052 | <p>An important theorem here : </p>
<blockquote>
<p>If $f'$ is continuous and defined on an interval I, then $f$ is also defined and continuous on I.</p>
</blockquote>
<p>Using this theorem, we get $f'(x)\frac{p}{2\sqrt{x}}$. $f'$ is not defined only for $x\in\Bbb R_-$. So $f$ is continuous for $x>0$, therefore ... |
66,314 | <p>This is very similar to my earlier question <a href="https://mathematica.stackexchange.com/questions/60069/one-to-many-lists-merge">One to Many Lists Merge</a> but somehow different. I have two lists, first column in each list represents its key. I want to merge these two lists. The only problem is that these two l... | andre314 | 5,467 | <pre><code>rules1 = Append[#[[1]] -> Rest[#] & /@ list1, _ -> {-99.99, -99.99}];
rules2 = Append[#[[1]] -> Rest[#] & /@ list2, _ -> {-99.99, -99.99, -99.99}];
Table[Join[{i}, i /. rules1, i /. rules2], {i,Union[First /@ list1, First /@ list2]}]
</code></pre>
<blockquote>
<p>{{1, a, aa, 10, 100, 1... |
3,267,550 | <p>Considering the system <span class="math-container">$x_{k+1}=Ax_k+Bu_k$</span> with quadratic cost </p>
<p><span class="math-container">$J^* = \min x_N^T S x_N + \sum_{k=0}^{N-1} x_k^T Qx_k+u^T_kRu_k$</span></p>
<p>where <span class="math-container">$Q,S\succeq 0, R\succ 0$</span>. The optimal state feedback is fo... | JMJ | 295,405 | <p>If <span class="math-container">$A = B$</span> then it's trivially true that <span class="math-container">$A - B \geq 0$</span>. </p>
|
1,380,819 | <p>Hi: I'm reading some introductory notes on hilbert spaces and there is a step in a proof that I don't follow. I will put the exact statement below. If someone could explain how it is obtained, it's appreciated. Note that commans between two terms when they have < and > around them denotes the innner product. Also... | Archaick | 191,173 | <p>For clarity's sake, I've rewritten this second to last line as</p>
<p>$$\left< x, x - \sum_{n=1}^{m}\left[ <x,e_{n}>e_{n}\right] \right>
- \sum_{n=1}^{m}\left[ <x, e_{n}> \left < e_{n}, x - \sum_{i=1}^{m} <x,e_{i}>e_{i} \right >\right].$$
By linearity of the inner-product,
$$\left<... |
3,200,940 | <p><a href="https://i.stack.imgur.com/7k9P8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7k9P8.jpg" alt="enter image description here"></a></p>
<p>Velleman's logic in sentence 3 under figure 4 is confusing me. He is using lines two and four of the truth table to infer what Q of line 1 should be. ... | lemontree | 344,246 | <p>Attmittedly it took me some time too to figure out what the author meant; here's what I think is intended: </p>
<p>An inference is valid iff in all the rows where all of the premises are true, the conclusion is true as well, i.o.w., there is no row where all premises are true but the conclusion is false. To check ... |
921,144 | <p>Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the... | Lehs | 171,248 | <p>Continuity is a condition on the function for given single points in the domain, while uniform continuity is a condition on the function for given pairs of points in the domain. Often one is sloppy with the quantifiers, but conditions of uniform continuity should start with two $\forall x_1 \forall x_2$ and then the... |
3,830,636 | <p>This is something I'm doing for a video game so may see some nonsense in the examples I provide.</p>
<p>Here's the problem:
I want to get a specific amount minerals, to get this minerals I need to refine ore. There are various kinds of ore, and each of them provide different amounts of minerals per volume. So I want... | Sil | 290,240 | <p>Rob's answer shows the problem definition in both linear and integer programming. Here is an integer programming code for experimenting. In practice you would typically use some library for the job, here is an example in Python 3 using <a href="https://docs.python-mip.com/en/latest/quickstart.html" rel="nofollow nor... |
2,615,626 | <p>The problem I have is:</p>
<blockquote>
<p>$\lim \limits_{x \to \infty} \sin{(x)}\ e^{-x}$</p>
</blockquote>
<p>Things I've tried:</p>
<ol>
<li><p>Researching how to do this problem, I've come across kind of similar examples that use either Taylors Rule, L'Hopitals Rule, or the Squeeze Theorem. Not sure which o... | Peter Szilas | 408,605 | <p>For fun:</p>
<p>Let $x \gt 0:$</p>
<p>$0\le |\dfrac{\sin x}{e^x}| \le \dfrac{1}{e^x}\lt\dfrac{1}{x}.$</p>
<p>Used: $e^x = 1+x+x^2/2! +...\gt x.$</p>
<p>Let $\epsilon >0$ be given. </p>
<p>Choose $M$, real, such that $M >1/\epsilon.$</p>
<p>For $x\gt M$ we have:</p>
<p>$0\le |\dfrac{\sin x}{e^x}| \lt \df... |
3,754,819 | <p>Evaluate the integral:
<span class="math-container">$$\int_{1}^{\sqrt{2}} \frac{x^4}{(x^2-1)^2+1}\,dx$$</span></p>
<p>The denominator is irreducible, if I want to factorize and use partial fractions, it has to be in complex numbers and then as an indefinite integral, we get
<span class="math-container">$$x + \frac{\... | Nikunj | 287,774 | <p>Start by writing <span class="math-container">$x^4 = (x^2 - 1 + 1)^2$</span>
<span class="math-container">$\implies x^4 = (x^2-1)^2 + 1 + 2(x^2-1)$</span></p>
<p>So the our integral becomes:</p>
<p><span class="math-container">$$\int_1^{\sqrt2}\frac{(x^2-1)^2 + 1 + 2(x^2-1)}{(x^2-1)^2 + 1}\,dx$$</span>
<span class="... |
1,072,524 | <p>Suppose $X$ is an integral scheme. I would like to show that the restriction maps
$res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could give me some assistance with this. This is exercise 5.2 I on Ravi Vakil's notes. Thank you!</p>
| Amitai Yuval | 166,201 | <p>In other words, we want to show that a section $s\in O_X(U)$ that vanishes on $V$ is necessarily the trivial section. Since $X$ is integral, $U$ is irreducible, thus $V$ is dense in $U$, and we're done.</p>
|
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Thierry Zell | 8,212 | <p>I've already given my opinion, and this is more of a remark: how the pros and cons are weighed between blackboard and slides should be influenced by a whole collection of classroom factors, and the first one among them should probably be class size.</p>
<p>This is a rather obvious remark, but I thought it was worth... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Andrew Stacey | 45 | <p>I'm going to try to answer the actual question rather than saying whether I think that chalk or projector is better. That "question" being:</p>
<blockquote>
<p>It would very much like to hear about your experiences with using slides in the classroom, possible pitfalls that you may have notices, and ways you have... |
2,432,817 | <p>Let $X$ and $Y$ be topological spaces and let $A \subseteq X$ be a subspace of $X$. Suppose $A$ is homeomorphic to some subspace $B \subseteq Y$ of $Y$. Let $f$ explicitly denote this homeomorphism.</p>
<p>If $f : A \to B$ is a homeomorphism, does $f$ extend to a homeomorphism between $\text{Cl}_X(A)$ and $\text{Cl... | egreg | 62,967 | <p>Let $A$ be a non closed subset of the compact space Hausdorff space $Y$, with the induced topology; now take $X=A$ and $f\colon X\to Y$ the inclusion map. The closure of $A$ in $X$ is $A$, not compact; the closure of $f(A)$ in $Y$ is compact.</p>
<p>Explicit example: $A=X=(0,1)$, $Y=[0,1]$.</p>
|
915,414 | <p>I recently did some work to try to find $\int{\frac{dx}{Ax^3 - B}}$, but I'm always paranoid that my solution has some minor trivial error in the middle of the process that screwed up the end result entirely, so could someone please help check my solution?</p>
<p>The first step to my solution is to eliminate $A$ an... | Claude Leibovici | 82,404 | <p>I think that we could make this simpler without involving complex numbers. </p>
<p>Just as rogerl did using partial fractions, we have $$\frac{1}{u^3-1} =\frac{1}{3}\frac{1}{u-1}-\frac{1}{3}\frac{u+2}{u^2+u+1}$$ Now $$\frac{u+2}{u^2+u+1}=\frac{1}{2}\frac{2u+4}{u^2+u+1}=\frac{1}{2}\Big(\frac{2u+1}{u^2+u+1}+\frac{3}{... |
915,016 | <p>Let $S$ be a non - empty set and $F$ be a field. Let $C(S,F)$ denote the set of all functions $f\in \mathcal F(S,F)$ such that $f(s)=0$ for all but a finite number of elements of $S$. Prove that $C(S,F)$ is a subspace of $\mathcal F(S,F)$</p>
<p>My progress:
I have shown that if $f,g\in C(S,F)$ and $a\in F$ then $f... | Dmoreno | 121,008 | <p>Why to learn that <em>complicated</em> formula? Will you even remember it when taking your exam? Why not using the <em>method of characteristics</em>, which I think it should be firstly taught when learning PDEs. In your case, it reads:</p>
<p>$$\frac{\mathrm{d}t}{1} = \frac{\mathrm{d}x}{1} = \frac{\mathrm{d}u}{(x+... |
1,535,376 | <p>I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)</p>
| JP McCarthy | 19,352 | <p>Gaussian Elimination can be used as long as you are not using decimal rounding.</p>
<p>If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.</p>
<p>I don't know without a Google when complete pivoting is necessary.</p>
|
1,535,376 | <p>I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)</p>
| eepperly16 | 239,046 | <p><strong>Rule of Thumb/TL;DR:</strong> When doing calculations using floating point numbers (such as double, single, and float data types in many common programming languages), use partial pivoting unless you know you're safe without it and complete pivoting only when you know that you need it.</p>
<hr>
<p><strong>... |
1,534,246 | <p>I'm trying to simplify this boolean expression:</p>
<p>$$(AB)+(A'C)+(BC)$$</p>
<p>I'm told by every calculator online that this would be logically equivalent:</p>
<p>$(AB)+(A'C)$</p>
<p>But so far, following the rules of boolean algebra, the best that I could get to was this: </p>
<p>$(B+A')(B+C)(A+C)$</p>
<p>... | Graham Kemp | 135,106 | <p>Well, clearly there's either $A$ or $A'$ in the first two terms, so use this to split the third wheel up, and absorb the pieces.</p>
<p>$\begin{align}(AB)+(A'C)+(BC) & = (AB)+(A'C)+(A+A')(BC) \\ & = (AB)+(A'C)+(ABC)+(A'BC) \\ & = (AB+ABC)+(A'C+A'BC) \\ & = (AB)(1+C)+(A'C)(1+B) \\ & = (AB)+(A'C)\... |
1,049,933 | <p>If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? </p>
<p>In my intuition, periodicity would correspond to an $-1$-eigenvalue of $M$, but I don't know if that is true or how to formalize it.</... | Woett | 65,418 | <p>By the most recent bound on <a href="https://en.wikipedia.org/wiki/Linnik%27s_theorem" rel="nofollow noreferrer">Linnik's Theorem</a>, there is an absolute constant <span class="math-container">$c$</span> such that for every prime <span class="math-container">$q < cp_n^{1/5}$</span>, there is a prime <span class=... |
4,242,093 | <p><em><strong>Question:</strong></em></p>
<blockquote>
<p>Let <span class="math-container">$G=(V_n,E_n)$</span> such that:</p>
<ul>
<li>G's vertices are words over <span class="math-container">$\sigma=\{a,b,c,d\}$</span> with length of <span class="math-container">$n$</span>, such that there aren't two adjacent equal ... | Mike Desgrottes | 464,046 | <p>Claim: For <span class="math-container">$n > 3$</span>, the graph <span class="math-container">$G = (V_{n},E_{n})$</span> has no Euler cycle.</p>
<p>When <span class="math-container">$n$</span> is even, we consider the string <span class="math-container">$w = a_{1}a_{2}...a_{n}$</span> with <span class="math-cont... |
1,977,736 | <p>I have these notations in an exercise and I can't understand them, the exercise is in French and I tried to translate it to English.</p>
<p><strong>("e" is the neutral element, "*" is a law of composition)</strong> (?)</p>
<p>1) Let (G,*) a group such as: </p>
<p>∀x ∈ G, x²= e</p>
<p><strong>-Is x² <=> x*x ?<... | ArtW | 191,773 | <p>Here is a sketch of a possible proof.</p>
<p>Suppose $f$ is a Möbius transformation taking $\mathbb{R}_{\infty}$ to $\mathbb{R}_{\infty}$. Then I claim the coefficients $a,b,c,d$ of $f$ are all real. suppose $a_1,a_2,a_3\in\mathbb{R_{\infty}}$ are three distinct points that are mapped to $b_1,b_2,b_3\in\mathbb{R_{\... |
1,977,736 | <p>I have these notations in an exercise and I can't understand them, the exercise is in French and I tried to translate it to English.</p>
<p><strong>("e" is the neutral element, "*" is a law of composition)</strong> (?)</p>
<p>1) Let (G,*) a group such as: </p>
<p>∀x ∈ G, x²= e</p>
<p><strong>-Is x² <=> x*x ?<... | Nitin Uniyal | 246,221 | <p>$w=\frac{az+b}{cz+d}$ gives $z=\frac{b-wd}{cw-a}$. The upper half plane is $y\geq 0$ or $z-\overline z\geq 0$</p>
<p>$\implies \frac{b-wd}{cw-a}-\frac{b-\overline wd}{c\overline w-a}\geq 0$</p>
<p>$\implies (ad-bc)(w-\overline w)\geq 0$ (On simplifying)</p>
<p>You need $ad-bc=1$ so that $w-\overline w\geq 0$ is t... |
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| bof | 111,012 | <p>If $n\gt1$ then $\sqrt[n]{n!}$ is not an integer (so it is an irrational number). A <a href="https://math.stackexchange.com/questions/1930933/is-sqrtnn-ever-an-integer/1930938#1930938">proof</a> using <a href="https://en.wikipedia.org/wiki/Bertrand%27s_postulate" rel="nofollow noreferrer">Bertrand's postulate</a> ha... |
121,784 | <p>I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.</p>
<p>You start with the <a href="http://mathworld.wolfram.com/RhombicDodecahedron.html" rel="nofollow noreferrer">rhombic dodecahedron</a>, subdivide it into four parallellepipeds, <a href="https://i.stack.img... | André Henriques | 5,690 | <p>It's the <a href="http://en.wikipedia.org/wiki/Minkowski_addition" rel="noreferrer">Minkowski sum</a> of the rhombic dodecahedron with a regular tetrahedron.
(The rhombic dodecahedron is it self the Minkowski sum of four segments).</p>
|
1,030,050 | <p>Assignment:</p>
<blockquote>
<p>Let $f$ be Lebesgue - measurable and $a,b \in \mathbb{R}$ with the property:
$$\frac{1}{\lambda(M)} \cdot \int_Mf\ d\lambda \in [a,b]$$
for all Lebesgue - measurable sets $M \subset \mathbb{R}^n$ with $0 < \lambda(M) < \infty$.</p>
<p>Show that: $f(x) \in [a,b]$ almo... | Chazz | 92,822 | <p>Since $[a,b]^{c}$ is open we can write it as a countable union of disjoint intervals. So it is enough to prove that $\lambda(f^{-1}(I))=0$, where $I:={B(x,r)}\subseteq [a,b]^{c}$. Write $E:=f^{-1}(I)$.</p>
<p>If $\lambda(E)>0$, then</p>
<p>$$\left|\frac{1}{\lambda(E)}\int_{E}f d\lambda-x\right|\leq\frac{1}{\lam... |
220,618 | <p>The cyclic group of $\mathbb{C}- \{ 0\}$ of complex numbers under multiplication generated by $(1+i)/\sqrt{2}$</p>
<p>I just wrote that this is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$ making a polar angle of $\pi/4$. I am not sure what to do next. My book say there are 8 elements. </p>
<p>Working backwards, maybe... | wj32 | 35,914 | <p>Firstly, $\mathbb{C}-\{0\}$ is not a cyclic group generated by that complex number (was that a typo?). The element $1/\sqrt{2}(1+i)=e^{\pi i/4}$ generates a cyclic <strong>subgroup</strong> of $\mathbb{C}-\{0\}$, so you need to find the smallest positive $k$ for which $(e^{\pi i/4})^k=e^{k\pi i/4}=1$ (since $1$ is t... |
720,794 | <p>Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and </p>
<p>$$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$
$$C = B + 1$$
$$H = G + 3$$</p>
<p>find (edit: without a calculator) $A,B,C,D,E,F,G,H,J$</p>
<p>I could only deduce that $D\ge E$, from the first one. Elim... | Guy | 127,574 | <p>Too long for a comment.</p>
<p>Testing following python code(brute forcing on finite sets is acceptable I think)</p>
<pre><code>def funct(t):
a,b,c,d,e,f,g,h,j=t
return (a/j)*(pow(b+c,d-e)-pow(f,g*h))
import itertools
for i in itertools.permutations(range(1,10))
if funct(i)==10:
prin... |
720,794 | <p>Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and </p>
<p>$$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$
$$C = B + 1$$
$$H = G + 3$$</p>
<p>find (edit: without a calculator) $A,B,C,D,E,F,G,H,J$</p>
<p>I could only deduce that $D\ge E$, from the first one. Elim... | ml0105 | 135,298 | <p>We have that $J|A$, and that $\frac{A}{J}|10$. So let's first consider the possible divisors of $10$, which are $1, 2, 5$. Clearly, $\frac{A}{J} \neq 5$ is the most likely option, based on the possible values. So $\frac{A}{J} = 2$. How many ways can we get this? Consider pairs $(A, J)$. We have $(2, 1)$, $(4, 2)$, $... |
720,794 | <p>Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and </p>
<p>$$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$
$$C = B + 1$$
$$H = G + 3$$</p>
<p>find (edit: without a calculator) $A,B,C,D,E,F,G,H,J$</p>
<p>I could only deduce that $D\ge E$, from the first one. Elim... | Thanos Darkadakis | 105,049 | <p>This is how I would find the solution with no computer aid:</p>
<p>You can write the equation as: $$(B+C)^{D-E} - F^{GH} = 10\frac{J}{A} $$
You can see that the RHS can take very small values ($\le90$). So, I would start with the assumption that it is unlikely to have two very large powers that have difference $\le... |
1,946,637 | <p>I have come across this integral and have been unable to solve it so far.</p>
<p>$$I=\int_{-\infty}^{0.29881}e^{-0.5x^2}\,\mathrm dx$$</p>
| E.H.E | 187,799 | <p>$$\frac{2^n(z^2)^n}{3^n+4^n}=\frac{2^n(z^2)^n}{4^n\left(1+(\frac{3}{4})^n\right)}=\frac{z^{2n}}{2^n\left(1+(\frac{3}{4})^n\right)}$$
now take the n-root
$$L=\lim_{n\rightarrow \infty }\left(\frac{z^{2n}}{2^n\left(1+(\frac{3}{4})^n\right)}\right)^{\frac{1}{n}}=\lim_{n\rightarrow \infty }\frac{z^2}{2(1+(\frac{3}{4})^n... |
1,003,020 | <p>Without recourse to Dirichlet's theorem, of course.
We're going to go over the problems in class but I'd prefer to know the answer today.</p>
<p>Let $S = \{3n+2 \in \mathbb P: n \in \mathbb N_{\ge 1}\}$</p>
<p>edit:</p>
<p>The original question is "the set of all primes of the form $3n + 2$, but I was only consid... | vadim123 | 73,324 | <p>The general strategy is to find a (large) number $n$ that is relatively prime to each of the existing list of such primes, and is also congruent to 2 modulo 3. The prime factorization of $n$ cannot consist only of primes congruent to $1$ modulo $3$, since the product of any number of such is still $1$ modulo $3$. ... |
465,945 | <p>I just need some verification on finding the basis for column spaces and row spaces.</p>
<p>If I'm given a matrix A and asked to find a basis for the row space, is the following method correct?</p>
<p>-Reduce to row echelon form. The rows with leading 1's will be the basis vectors for the row space.</p>
<p>When l... | Chris Tang | 695,705 | <p>Your procedure is correct, and I'd like to summary them:</p>
<h2>Finding Row Space</h2>
<ul>
<li>Strategy: do Gaussian elimination on the matrix M; non-zero rows of reduced matrix M' form the set spanning row space. </li>
<li>Reason: 1) elementary row operations don't change the row space of M, since row vectors a... |
2,000,556 | <p>Suppose $N>m$, denote the number of ways to be $W(N,m)$</p>
<h3>First method</h3>
<p>Take $m$ balls out of $N$, put one ball at each bucket. Then every ball of the left the $N-m$ balls can be freely put into $m$ bucket. Thus we have: $W(N,m)=m^{N-m}$.</p>
<h3>Second method</h3>
<p>When we are going to put $N$... | Brian M. Scott | 12,042 | <p><strong>hypergeometric</strong> has given a good analysis of the problem for indistinguishable balls.</p>
<p>When the balls are distinct, we can number them $1$ through $N$. If we number the buckets $1$ through $m$, each assignment of balls to buckets with at least one ball in each bucket can be thought of as a fun... |
66,719 | <p>I know some elementary proofs of this fact. I was wondering if there's some short slick proof of this fact using the structure of the $2$-adic integers? I'm looking for a proof of this fact that's easy to remember.</p>
| Jyrki Lahtonen | 11,619 | <p>The shortest proof I can think of goes as follows. Hensel is not needed. A counting argument will suffice.</p>
<p>We take as the starting point that the group of units $U_n$ of the ring $\mathbf{Z}/2^n\mathbf{Z}$ has the structure $U_n\simeq C_2\times C_{2^{n-2}}$. Therefore exactly one quarter of elements of $U_n$... |
301,724 | <p>There's an example in my textbook about cancellation error that I'm not totally getting. It says that with a $5$ digit decimal arithmetic, $100001$ cannot be represented.</p>
<p>I think that's because when you try to represent it you get $1*10^5$, which is $100000$. However it goes on to say that when $100001$ is r... | Ross Millikan | 1,827 | <p>Yes, you only have five decimal digits available. $100001=1.00001*10^5$ but I have six digits in the mantissa. Clearly it is closer to go to $1.0000$ than to $1.0001$, so that is what we will do. So the numbers around here that can be represented are $99998, 99999, 100000, 100010, 100020,$ etc.</p>
|
532,525 | <p>Is $P( A \cup B \,|\, C)$ the same as $P(A | C) + P(B | C)$ ?
Here $A$ and $B$ are mutually exclusive.</p>
| Henno Brandsma | 4,280 | <p>One could use nets as well (if you know them; they're pretty popular in functional analysis as a generalisation of sequences that work in all topological spaces): Take any $z \in \operatorname{cl}(A) + \operatorname{cl}(B)$, so $z= x + y$ with $x \in \operatorname{cl}(A)$, $y \in \operatorname{cl}(B)$, so we have a ... |
651,731 | <blockquote>
<p>Let $\{f_n\}$be sequence of bounded real valued functions on $[0,1]$ converging at all points of this interval. Then
If $\int^1_0 |f_n(t)-f(t)|dt\, \to 0$ as $n \to \infty$,does $\lim_{n \to \infty} \int^1_0 f_n(t) dt\,=\int^1_0 f(t)dt\,$</p>
</blockquote>
<p>I just know that if somehow we can sho... | TheNumber23 | 90,401 | <p>This result is false. Let $f_{n}$ be zero for $x\leq\tfrac{1}{n+1}$ and $0$ for $x\geq \tfrac{1}{n}$. Then connect both points $(\tfrac{1}{n+1},0)$ and $(\tfrac{1}{n},0)$ to $(\tfrac{1}{2n}+\tfrac{1}{2n+1},n(n+1))$, Notice what the integral for each $f_{n}$ is and see what it converges to. </p>
|
3,065,331 | <p>In <a href="https://rads.stackoverflow.com/amzn/click/0199208255" rel="nofollow noreferrer">Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers</a> one of the very first stated equations are, as in the title of the question,</p>
<p><span class="math-container">$$
\ddot{x} = \fr... | J.G. | 56,861 | <p>By the chain rule, <span class="math-container">$$\frac{d}{dx}\left(\frac12\dot{x}^2\right)=\frac{1}{\dot{x}}\frac{d}{dt}\left(\frac12\dot{x}^2\right)=\frac{1}{\dot{x}}\times\dot{x}\ddot{x}=\ddot{x}.$$</span></p>
|
2,555,200 | <p>Let $l$ be the smallest positive linear combination of $a,b\in \mathbb{Z}^+$ i.e.,$$l := \min\{ax+by >0 : x,y\in\mathbb{Z}\}.$$
Now, according to @Brahadeesh's answer here, <a href="https://math.stackexchange.com/questions/2553324/proof-for-gcd-being-the-smallest-linear-combination-of-a-b-in-mathbb-z">Proof for $... | user7530 | 7,530 | <p>I’m not sure that there’s much difference between what you want and the “direct” proof. I would argue as follows. Suppose for contradiction that $k\vert a$ and $k\vert b$. Then $k\vert l$, and by an identical argument $l\vert k$. Therefore $k=l \not> l,$ a contradiction.</p>
|
1,423,728 | <p>The definition of a limit in <a href="http://rads.stackoverflow.com/amzn/click/0321888545" rel="nofollow noreferrer">this book</a> stated like this </p>
<p><a href="https://i.stack.imgur.com/tKjaa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tKjaa.png" alt="enter image description here"></a><... | Subhasish Basak | 266,148 | <p>Suppose there is no open interval around $x_0$, where $f(x)$ is defined for all $x$ in that interval.</p>
<p>Which means for every $\delta >0$ $\exists x_1 \in (x_0-\delta,x_0+\delta)$ such that $f(x_1)$ is not defined. So we cannot say $|f(x)-L|<\epsilon$ $\forall x\in(x_0-\delta,x_0+\delta)$ , as you can n... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Michael Hardy | 6,316 | <p><s>Here's the math WikiProject's "Current activity" page, which lists each day's new articles, articles for which deletion is proposed, articles needing expert attention, articles needing references, and articles needing various other sorts of work: <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathem... |
2,156,606 | <p>I am stuck in this exercise of calculus about solving this indefinite integral, so I would like some help from your part: </p>
<blockquote>
<p>$$\int \frac{dx}{(1+x^{2})^{\frac{3}{2}}}$$</p>
</blockquote>
| Simply Beautiful Art | 272,831 | <p>We have $t>0$.</p>
<p>Recall that</p>
<p>$$\int\frac1{(t^2-x^2)^{1/2}}\ dx=\arcsin\left(\frac xt\right)+c_1$$</p>
<p>Take the derivative of both sides with respect to $t$ to get</p>
<p>$$\int\frac{-t}{(t^2-x^2)^{3/2}}\ dx=\frac{-x}{|t|\sqrt{t^2-x^2}}+c_2$$</p>
<p>Setting $t=1$ and simplifying, we reach</p>
... |
57,642 | <p>I'm looking for the shortest and the clearest proof for this following theorem:</p>
<p>For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , I need to show $V= \ker T^n \oplus $ Im $T^n$.</p>
<p>Any hints? I don't know where to start from.</p>
<p>Thank you.</p>
| jspecter | 11,844 | <p>Let $E_1,...,E_i$ be the generalized Eigenspaces associated to $T.$ Show by examining the Jordan canonical form of $T$ that $T^n$ acts on $E_j$ as an automorphism if the eigenvalue associated to $E_j,$ denoted $\lambda(E_j),$ is non-zero and as the zero transformation otherwise. It follows $$\mathrm{Ker } T^n = \dis... |
57,642 | <p>I'm looking for the shortest and the clearest proof for this following theorem:</p>
<p>For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , I need to show $V= \ker T^n \oplus $ Im $T^n$.</p>
<p>Any hints? I don't know where to start from.</p>
<p>Thank you.</p>
| Pierre-Yves Gaillard | 660 | <p>If $S$ is an endomorphism of a finite dimensional vector space $V$, then
$$V=\mathrm{Ker}\ S\ \oplus\ \mathrm{Im}\ S\Leftrightarrow\mathrm{Ker}\ S\ \cap\ \mathrm{Im}\ S=0\Leftrightarrow\mathrm{Ker}\ S^2=\mathrm{Ker}\ S.$$ </p>
<p>Put $S:=T^n$. </p>
<hr>
<p>I find Robert Israel's argument wonderful, but it seems ... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Aryabhata | 1,102 | <p>The odd town puzzle.</p>
<p>You have a town with $m$ clubs formed by $n$ citizens of the town.</p>
<p>The clubs are so formed that</p>
<ul>
<li>Each club has an odd number of members.</li>
<li>Any two clubs have an even number of common members. (Could be zero too).</li>
</ul>
<p>Show that $m \le n$.</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Fixee | 7,162 | <p>Prove that any 2-coloring of a $K_6$ has <strong>two</strong> monochromatic $K_3$'s.</p>
|
3,255,174 | <p>How can I find the volume bounded between <span class="math-container">$z=(x^2+y^2)^2$</span> and <span class="math-container">$z=x$</span>? </p>
<p>My idea so far is to use cylindrical polar coordinates and <span class="math-container">$z$</span> limit is from <span class="math-container">$(x^2+y^2)^2$</span> to <... | José Carlos Santos | 446,262 | <p>In cylindrical coordinates, your equations become <span class="math-container">$z=r^4$</span> and <span class="math-container">$z=r\cos\theta$</span>. So, take <span class="math-container">$\theta\in\left[-\frac\pi2,\frac\pi2\right]$</span> (so that <span class="math-container">$\cos\theta\geqslant0$</span>). Now, <... |
387,519 | <p>The domain of the following function $$y=2$$ is just 2? And the image of it?</p>
<p>I don't think I quiet understand what the image of a function means. The domain is all values that it can assume, correct?</p>
<p>Could you please try to define the image of this equation too: $$y = 2x - 6$$ so I can try to underst... | AndreasT | 53,739 | <p>When you define a function you should always provide a domain (i.e. a start set) and a co-domain (i.e. a target set). Writing
$$
f:A\rightarrow B
$$
means that $A$ is the domain and $B$ the co-domain. $f$ is a function if it "associates" every single element $a\in A$ with a unique element $b\in B$. We call such $b... |
2,309,418 | <p>How can I integrate this function?</p>
<p>$$\int \left( {3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}\right)dx$$</p>
<p>Using a previous example I have found:</p>
<p>$$\int \left({3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}dx\right) ={A\over x^2+1}+{B \over x+1}+{Cx+D\over (x+1)^2}$$</p>
<p>And then:
$$\int 3x^3+3x^2+3x+1 ={A(... | Ahmed | 126,745 | <p><strong>Hint</strong>
Instead of what you wrote, you have to take:
$$\int \left({3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}dx\right) ={Ax+B\over x^2+1}+{C \over x+1}+{D\over (x+1)^2}$$
Since $x^2+1$ is an erriducible polynomial.</p>
|
594,811 | <p>This is my first post, sorry for my naiveness..</p>
<p>I know a basic equation that relates Gram-schmidt matrix and Euclidean distance matrix:</p>
<p>$XX'=-0.5*(I-J/n)*D*(I-J/n)'$</p>
<p>Where $X$ is centered data (is $d \times n$), $I$ is identity matrix, $J$ is a matrix filled with ones (1), $n$ is the number o... | lynne | 116,783 | <p>The solution to $A.X.B = Y$ in a least-squares sense is
$$ X = A^+.Y.B^+ + W - A^+.A.W.B.B^+ $$
where $W$ is arbitrary and $A^+$ denotes the pseudo-inverse of $A$.</p>
<p>In your problem, $A=B$ equals the centering matrix (see <a href="http://en.wikipedia.org/wiki/Centering_matrix" rel="nofollow">http://en.wikiped... |
1,790,222 | <p>I know that $[0,1]$ and a unit circle $\mathbb{S}^1$ are one-point compactifications of $\mathbb{R}$ under some suitable homeomorphism. But how does one construct the Stone–Čech compactification? </p>
| egreg | 62,967 | <p>The Stone-Čech compactification of $\mathbb{R}$ can be functorially built as the maximal spectrum of $C_b(\mathbb{R})$, the ring of bounded continuous real functions on $\mathbb{R}$.</p>
<p>The maximal spectrum $\operatorname{Max}(R)$ of a commutative ring is the set of all maximal ideals, with the spectral topolog... |
42,258 | <p>I have a Table of values e.g. </p>
<pre><code>{{x,y,z},{x,y,z},{x,y,z}…}
</code></pre>
<p>How do I replace the the "z" column with a List of values?</p>
| Z-Y.L | 4,733 | <p>Actually $\partial _t f[t]$ in Mathematica is interpreted as <code>D[f[t],t]</code> by default. You don't need to redefine it. </p>
<p>Considering $\partial _t = D[f,t]$ given by the OP is only an example of what the OP wants to do, I regard this question as a way to redefine the basic rules for the input of the ex... |
2,703,323 | <p>How can one show that the limit of the following is $1$?</p>
<p>$$\lim_{x\to 0}\frac{\frac{1}{1-x}-1}{x}=1$$</p>
| ervx | 325,617 | <p>Hint:</p>
<p>$$
\frac{\frac{1}{1-x}-1}{x}=\frac{\frac{1-(1-x)}{1-x}}{x}=\frac{x}{x(1-x)}=\frac{1}{1-x}.
$$</p>
|
4,196,125 | <p>Can someone tell me where this calculation goes wrong?
I get (2 3 4)(1 2 3 4 5 6)^-1 = (1 6 5 2).
My book and Mathematica get (1 6 5 4).
I have read several explanations of how to multiply permutations in cycle notation and have
worked dozens of examples successfully, but I always get this one wrong.</p>
<p>(2 3 4)(... | Narasimham | 95,860 | <p>HINT</p>
<p>It is also the same as the volume obtained by revolving circle (with Shifing / Reflection)</p>
<p><span class="math-container">$$ (x-2)^2+y^2= 1$$</span></p>
<p>about the y-axis.</p>
|
1,593,339 | <p>What is $y$ in $$J^\frac{1}{2}f(x)=y$$
$$f(x)=w\sin(x)$$
where $w$ is a constant?</p>
| Simply Beautiful Art | 272,831 | <p>Allow $\frac{d^n}{dx^n}e^{cx}=c^ne^{cx}$. This should be fairly obvious and holds by induction.</p>
<p>Now let $c=i$ and we can get Euler's formula to do all the work.</p>
<p>$$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$</p>
<p>$$\frac{d^n}{dx^n}\sin(x)=\frac{d^n}{dx^n}\frac{e^{ix}-e^{-ix}}{2i}$$</p>
<p>$$=\frac{i^ne^{... |
114,898 | <p>A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers. Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant. Why is this? Without some kind of vanishing of cohomology or information on the base, I don't see why this is true. <... | Alexandre Eremenko | 25,510 | <p>Here it is:
MR0015154
Salem, R.; Zygmund, A.
Lacunary power series and Peano curves.
Duke Math. J. 12, (1945). 569–578. </p>
|
416,387 | <p>In the process of solving a DE and imposing the initial condition I came up with the following question.</p>
<p>I've reached the stage that</p>
<p>$$\ln y + C = \int\left(\frac{2}{x+2}-\frac{1}{x+1}\right)dx$$
$$\Rightarrow \ln y +C=2\ln|x+2|-\ln|x+1|$$
$$\Rightarrow y=A\frac{(x+2)^2}{|x+1|}.$$</p>
<p>Now I had a... | Abhra Abir Kundu | 48,639 | <p>$\ln (-x)$ does not make sense for $x>0$ so modulus sign must always be there for the function to make sense. You can get rid of modulus sign iff you know that $\ln(x)$ is used for some $x$ greater than zero(i.e. the parameter inside the function must be greater than zero.) </p>
<p>Like in this case if the modul... |
2,140,363 | <p>A fair die is rolled 10 times. Define N to be the number of distinct outcomes. For example, if we roll (3, 2, 2, 1, 4, 3, 1, 6, 2, 3) then N = 5 (the distinct outcomes being 1, 2, 3, 4 and 6). Find the mean and standard deviation of N.
Hint: Define $I_j =
\left\{
\begin{array}{ll}
1 & \text{if outcome j appea... | JMoravitz | 179,297 | <p><em>For the sake of brevity, I will assume that you already know the definitions of random variables, expected value, and standard deviation and will not formally define these terms. For a formal definition, seek out any textbook on the subject.</em></p>
<p>Our problem asks us to find the mean (<em>expected value<... |
2,140,363 | <p>A fair die is rolled 10 times. Define N to be the number of distinct outcomes. For example, if we roll (3, 2, 2, 1, 4, 3, 1, 6, 2, 3) then N = 5 (the distinct outcomes being 1, 2, 3, 4 and 6). Find the mean and standard deviation of N.
Hint: Define $I_j =
\left\{
\begin{array}{ll}
1 & \text{if outcome j appea... | Marko Riedel | 44,883 | <p>Supposing that the fair die has $n$ sides and is rolled $m$ times we
get for the probability of $k$ distinct outcomes by first principles
the closed form</p>
<p>$$\frac{1}{n^m} \times
m! [z^m] {n\choose k} (\exp(z)-1)^k.$$</p>
<p>Let us verify that this is a probability distribution. We obtain</p>
<p>$$\frac{... |
222,105 | <p>I want to use geometric shapes in Mathematica to build complex shapes and use my raytracing algorithm on it. I have a working example where we can get the intersections from a combination of a <code>Cone[]</code> and <code>Cuboid[]</code>, e.g </p>
<pre><code>shape1 = Cone[];
shape2 = Cuboid[];
(* add shapes in thi... | Tomi | 36,939 | <p>Tim Laska's solution is excellent. It is fast and accurate. However, for completeness, I have a solution for the <code>NDSolve</code> solution, where we can find the intersections instead of the (excellent) particle advancer (i.e. just jump between the intersections instead of advance). </p>
<p>By using the solutio... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.