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 |
|---|---|---|---|---|
3,716,439 | <p>Uhm I'm having trouble solving for x in these type of equations: <span class="math-container">$a^x + b^x = c^x$</span> (where x is the only unknown)</p>
<p>I saw an instance by Presh Talwalkar in MindYourDecisions. I tried solving for x and used an example with a known answer. <span class="math-container">$3^x + 4... | user577215664 | 475,762 | <p>Are you sure you copied correectly the system ? Maybe it's a minus sign instead of a positive one in one of the differential equation ?</p>
<p>Because if you substract both DE you get:</p>
<p><span class="math-container">$$x+3y=\sin t $$</span></p>
<p>Use this to solve the seond differential equation:
<span class="m... |
108,781 | <p>I want to add arrowheads and labels to the following plot. </p>
<pre><code>P1 = Plot[x, {x, 0, 2}, PlotStyle -> {Dashed, Red}, Filling -> Bottom];
P2 = Plot[6 x, {x, 0, 2}, PlotStyle -> {Dashed, Blue}];
Show[P1, P2, AspectRatio -> Automatic, Frame -> True,
PlotRangePadding -> None, AxesOrigin -... | kglr | 125 | <p>As an alternative to <code>Epilog</code>you can use the <code>PlotStyle</code> settings to add arrows: </p>
<pre><code>Plot[{x, 6 x}, {x, 0, 2}, Filling -> {1 -> Bottom},
PlotRange -> {0, 2}, AspectRatio -> 1, PlotRangePadding -> None,
AxesOrigin -> {0, 0}, Axes -> None, Frame -> True,
... |
1,645,211 | <p>The problem statement is,</p>
<p>Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$.</p>
<p>For (i),</p>
<p>I found out the basis of the null space for the system,</p>
<p>$$\begin{pmatrix}4&... | user311576 | 311,576 | <p>If $M(s)$ is supposed to be the $\sigma$-Algebra generated by $s$, then
it should suffice to use the definition of a measure and the general description of an element of $M(s)$.
If $M(s)$ is not said algebra, please clarify.</p>
|
1,371,921 | <p>The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$
and
$v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$
is a line. What is the equation of this line?</p>
<p>This is what I have so far: I set $v = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$. I was able ... | David Quinn | 187,299 | <p>There are two basic methods: either</p>
<ol>
<li>You can obtain the direction of the line by doing the cross-product of the normals $$\left(\begin {matrix}8\\1\\-12\end {matrix}\right)\times\left(\begin {matrix}6\\7\\-9\end {matrix}\right)=\left(\begin {matrix}75\\0\\50\end {matrix}\right) \text{which is parallel t... |
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | stewbasic | 197,161 | <p>Here is a proof that no such continuous functions exist. I'll use juxtaposition to denote function composition. As noted in the comments,
$$
h^2=fgh=f^2=ghf=g^2.
$$
Let $e=f^4=g^4=h^4$. Note that $fgf=hf=g$ and $gfg=gh=f$, so
$$
ef=g^4f=gf^2gf=gfg=f,
$$
$$
fe=fg^4=fgf^2g=gfg=f.
$$
Similarly $eg=g=ge$ and $eh=h... |
3,702,520 | <p>I've used cosine & sine rule, and after playing around with them, I'll always end up with an equation involving <span class="math-container">$\sin\theta,\cos\theta$</span> and other weird (and wonderful expressions). I've approached the question in many different ways: by showing <span class="math-container">$A... | AryanSonwatikar | 571,692 | <p><span class="math-container">$$x+2=-\sqrt{3}i$$</span>
Squaring,
<span class="math-container">$$x^2 +4x+4=-3$$</span>
Or,
<span class="math-container">$$x^2+4x+7=0\Rightarrow (x^2+4x+7)^2=0$$</span>
Open the brackets and use to simplify.</p>
|
3,702,520 | <p>I've used cosine & sine rule, and after playing around with them, I'll always end up with an equation involving <span class="math-container">$\sin\theta,\cos\theta$</span> and other weird (and wonderful expressions). I've approached the question in many different ways: by showing <span class="math-container">$A... | user376343 | 376,343 | <p>Real coefficients in the quartic inspired me to multiply <span class="math-container">$x + 2 + \sqrt{3}i\;$</span> by the conjugate, which gives</p>
<p><span class="math-container">$$(x + 2 + \sqrt{3}i)(x + 2 - \sqrt{3}i)=x^2+4x+7.$$</span></p>
<p>Dividing the given expression by <span class="math-container">$x^2+... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Mohammad Riazi-Kermani | 514,496 | <p>You may derive the equation for ellipse from the equation of circle by scaling your $x$ and $y$</p>
<p>$$ x^2 + y^2 = R^2 $$</p>
<p>Let $$x\to x/a,\text {and } y\to y/b$$</p>
<p>You get $$ (x/a)^2 + (y/b)^2 = R^2 $$</p>
<p>$$ (\frac {x}{aR})^2 + (\frac {y}{bR})^2 = 1 $$
$$ \frac {x^2}{A^2} + \frac {y^2}{B^2... |
26,074 | <p>Okay, so this was labeled as a "fun problem", but I'm having trouble knowing how to approach it.</p>
<p>I'm given: $\lim\limits_{x \to 0^+} f(x) = A$ and $\lim\limits_{x \to 0^-} f(x) = B$.</p>
<p>I need to find (or at least know where to start):</p>
<p>a) $\lim\limits_{x \to 0^+} f(x^3 - x)$</p>
<p>b) $\lim\lim... | Mariano Suárez-Álvarez | 274 | <p>When $x$ goes to zero, $x^3-x$ also goes to zero. But we can be more precise: in fact,</p>
<ul>
<li><p>when $x$ goes to zero from the left, $x^3-x$ goes to zero from the right, and</p></li>
<li><p>when $x$ goe to zero from the right, $x^3-x$ goes to zero from the left.</p></li>
</ul>
<p>Can you see how to use this... |
1,609,443 | <p>So my textbook states that it is $(-\infty,0)$ that is concave down, but why not $(-\infty,0]$? Could someone help?</p>
| Bob Krueger | 228,620 | <p>Let me give a more pedagogical explanation without referring to any specific definitions. This question is very similar to "Is $x^2$ increasing on $(0,\infty)$ or $[0,\infty)$?" Here is my answer to that question: if $x^2$ is increasing on $[0,\infty)$, then we would also have to agree by symmetry that $x^2$ is decr... |
1,609,443 | <p>So my textbook states that it is $(-\infty,0)$ that is concave down, but why not $(-\infty,0]$? Could someone help?</p>
| Jendrik Stelzner | 300,783 | <p>This depends on your definition of concavity. I prefer the following:</p>
<blockquote>
<p><strong><em>Definition:</em></strong> Let $I \subseteq \mathbb{R}$ be an intervall and $f$ a realvalued function defined on $I$. Then $f$ is <em>convex</em> on $I$ if for all $x,y \in I$ and $\lambda \in [0,1]$
$$
f(\lamb... |
171,697 | <blockquote>
<p>Let $f \in C^1(\mathbb R, \mathbb R)$ and suppose
$$ \tag{H}
\vert f(x) \vert\le \frac{1}{2}\vert x \vert + 3, \quad \forall x \in \mathbb R.
$$</p>
<p>Then, every solution of
$$
x'(t)+x(t)+f(x(t))=0, \quad t \in \mathbb R
$$
is bounded on $[0,+\infty]$. </p>
</blockquote>
<p>First of ... | copper.hat | 27,978 | <p>The goal is to show that the solution is bounded. (I have no idea what the $+\infty$ bit is about, it adds nothing as far as I can tell.)</p>
<p>We first need to show that the solution from any initial condition $x_0$ (at $t_0$) is defined on $[t_0,\infty)$. It is not immediately clear to me that 'we have global ex... |
2,840,425 | <p>I am trying to solve the assignment below with the quotient rule. Every time I get stuck on a similar assignment which has a square root inside the numerator of a fraction. What do I do with the square root in the numerator and how do I find the derivative?</p>
<p>$$\frac{\sqrt x+3}{x}$$</p>
| Peter Szilas | 408,605 | <p>An option:</p>
<p>$\dfrac{1}{ x^2-a^2}=\dfrac{1}{(x-a)(x+a)}=$</p>
<p>$\dfrac{1}{2a}[ \dfrac{1}{x-a} -\dfrac{1}{x+a}]$.</p>
|
2,873,636 | <p>I have been self studying mathematical logic through A Concise Introduction to Mathematical Logic by Wolfgang Rautenberg and got stuck. I am unable to understand how the value matrix corresponds with the truth table of a given operation for instance the AND operation. I know that in a matrix the place an element is ... | Rob Arthan | 23,171 | <p>The author is using an abbreviated form of the notation often used for binary operations on finite algebraic structures, leaving out the row and column labels. Drawn up in full, the table for conjunction would look like this:
$$
\begin{array}{c|cc}
{\land} & 1 & 0 \\\hline
1 & 1 & 0 \\
0 & 0 &... |
2,788,002 | <p>If I convert, say, #FF$1919$ to binary, I can do it in groups of three bytes like:</p>
<p>FF: $1111$ $1111$</p>
<p>$19$: $0001$ $1001$</p>
<p>$19$: $0001$ $1001$</p>
<p>So can I write that #FF$1919 =$ $1111$ $1111$ $0001$ $1001$ $0001$ $1001$?</p>
<p>Or do I have to write them as separate bytes like above?</p>
| mvw | 86,776 | <p>Yes, the positional systen works as expected:
$$
(FF1919)_{16}
= 15 \cdot 16^5 + 15 \cdot 16^4 + 1 \cdot 16^3 + 9 \cdot 16^2 + 1 \cdot 16^1 + 9 \cdot 16^0 \\
$$
Then we have
$$
16^k = (2^4)^k = 2^{4k}
$$
so
\begin{align}
(FF1919)_{16}
&=
(1111)_2 \cdot 2^{4\cdot 5} +
(1111)_2 \cdot 2^{4\cdot 4} +
(0001)_2 \cd... |
1,332,433 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to \mathbb R$</span> be such that for every sequence <span class="math-container">$(a_n)$</span> of real numbers,</p>
<p><span class="math-container">$\sum a_n$</span> converges <span class="math-container">$\implies \sum f(a_n)$</span> converges</p>
<p>Prov... | Martin Sleziak | 8,297 | <p>I have added an answer to collect various resources where this fact is proven. This answer is <a href="http://meta.math.stackexchange.com/tags/community-wiki/info">community-wiki</a> - feel free to add further references you are aware of.</p>
<p><strong>Papers:</strong></p>
<ul>
<li>Gerald Wildenberg: <em>Converge... |
33,778 | <p>Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.</p>
<p>I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in our final answer since we need... | Did | 6,179 | <p>Recall one of the most important characterizations of the exponential distribution:</p>
<blockquote>
<p>The random variable $Y$ is exponentially distributed with rate $\beta$ if and only if $P(Y\geqslant y)=\mathrm{e}^{-\beta y}$ for every $y\geqslant0$.</p>
</blockquote>
<p>Let $Z=X/Y$ and $t\gt0$. Conditioning... |
33,778 | <p>Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.</p>
<p>I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in our final answer since we need... | dohmatob | 168,758 | <p>Here is a one-line proof.
<span class="math-container">$$
\mathbb P(X/Y \le t) = \mathbb P(Y \ge X/t) = \mathbb E[\exp(-\beta X/t)] = \text{MGF}_X(-\beta/t) = (1 - (-\beta/t)1/\alpha)^{-1} = \frac{\alpha}{\alpha + \beta/t} = \frac{\alpha t}{\alpha t + \beta}.
$$</span></p>
<p><strong>N.B.:</strong> For the MGF of a... |
1,783,439 | <p>How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like?</p>
<p>The initial problem was "what is the dimension of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ ?</p>
... | MUH | 214,956 | <p>To prove that they are Linearly Independent, all we have to do is P.T. if
$$
x_1v_1+\cdots+x_{n-1}v_{n-1}=0\\
\Rightarrow x_1e_1-x_1e_n+x_2e_2-x_2e_n+\cdots+x_{n-1}e_{n-1}-x_{n-1}e_{n}=0\\
\Rightarrow x_1e_1+x_2e_2+\cdots+x_{n-1}e_{n-1}-(x_1e_n+x_2e_n+\cdots+x_{n-1}e_{n})=0 \\
\Rightarrow x_1=\cdots=x_{n-1}=0
$$</p... |
1,006,080 | <p>How to solve:
$$\left|\frac{x+4}{ax+2}\right| > \frac{1}{x}$$</p>
<p>What I have done:<br><br></p>
<p>I) $x < 0$:</p>
<p>Obviously this part of the inequation is $x\in(-\infty, 0), x\neq \frac{-2}{a}$<br><br></p>
<p>II) $x > 0$:</p>
<p>$$\left|\frac{x+4}{ax+2}\right| > \frac1x$$
$$\frac{|x+4|}{|ax+2... | Community | -1 | <p>This is not an answer, just an alternative exposition of your findings as far as I was able to reproduce them.</p>
<p>First, the video describes Archimedes' method for approximating $\pi$ via the following recurrence for the side length of a regular $3\cdot2^n$-gon inscribed in a circle, which I'll call $\ell_n$:
$... |
3,962,427 | <p>I have proven that the sequence is convergent.</p>
<p>Let <span class="math-container">$x_n=\frac{(2n)!!}{(2n+1)!!}$</span></p>
<p><span class="math-container">$\frac{x_{n+1}}{x_n}$</span> <span class="math-container">$=\frac{2n+2}{2n+3}<1$</span> Therefore the sequence decreases. On the other hand
<span class="m... | Varun Vejalla | 595,055 | <p>Similar to another answer, you have that <span class="math-container">$$x_n = \prod_{k=1}^{n}\frac{2k}{2k+1}$$</span></p>
<p>Then
<span class="math-container">$$\ln(x_n) = \sum_{k=1}^{n}\ln\left( \frac{2k}{2k+1} \right)$$</span></p>
<p>Each term will be negative and with the comparison test, you have that <span clas... |
1,025,139 | <p>How to prove the following? Should I use induction or something else?</p>
<blockquote>
<p>Let $n$ and $r$ be positive integers with $n \ge r$. Prove that
$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</p>
</blockquote>
<p><em>Attempted start:</em></p>
<p>Basis step: ${\binom{1}{... | Hypergeometricx | 168,053 | <p>Note that</p>
<p>$$\begin{align}
\binom ab+\binom a{b+1}&=\binom{a+1}{b+1}\\
\Rightarrow \binom ab&=\binom{a+1}{b+1}-\binom a{b+1}\end{align}$$</p>
<p>Hence
$$\begin{align}
\binom ir&=\binom {i+1}{r+1}-\binom i{r+1}\\
\sum_{i=r}^n\binom ir&=\sum_{i=r}^n\binom {i+1}{r+1}-\sum_{i=r}^n\binom i{r+1}\\... |
951,414 | <p>Consider the absolute value equation:</p>
<p>|x| + |x-2| +|x-4|= 6</p>
<p>How to find the solution(s)?</p>
<p>My attempt:</p>
<p>For |x|, we got x, for x>=0 and -x, for x <0</p>
<p>For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0</p>
<p>For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0</p>
<p>... | Alexander Vigodner | 92,276 | <p>You have 3 "nodes" for switching $|x|$ , $|x-2|$ and $x-4|$. The easiest way is to imagine real line with these points and split solution considering intervals $[-\infty,0],[0,2],[2,4],[4,\infty]$ separately. On each interval you can remove $||$ with certain sign and immediately solve the equation, checking that fin... |
951,414 | <p>Consider the absolute value equation:</p>
<p>|x| + |x-2| +|x-4|= 6</p>
<p>How to find the solution(s)?</p>
<p>My attempt:</p>
<p>For |x|, we got x, for x>=0 and -x, for x <0</p>
<p>For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0</p>
<p>For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0</p>
<p>... | drhab | 75,923 | <ul>
<li>$x<0$ leads to $6-3x=6$ </li>
</ul>
<p>Solve this equation and check wether the solution satisfies $x<0$. If so then you have found a solution of the original equation. If not then no solution exists that satisfies $x<0$.</p>
<p>Do the same for the cases:</p>
<ul>
<li><p>$0\leq x<2$</p></li>
<li... |
951,414 | <p>Consider the absolute value equation:</p>
<p>|x| + |x-2| +|x-4|= 6</p>
<p>How to find the solution(s)?</p>
<p>My attempt:</p>
<p>For |x|, we got x, for x>=0 and -x, for x <0</p>
<p>For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0</p>
<p>For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0</p>
<p>... | Mark Bennet | 2,906 | <p>Think of the modulus function measuring the distance from fixed points. So that $|x|=|x-0|$ is the distance from $0$.</p>
<p>So your sum measures the total distance from the points $0,2,4$</p>
<p>If $x$ is greater than $4$ the first two distances are greater than $4$ and $2$</p>
<p>If $x$ is less than $0$ we reac... |
1,483,191 | <p>It's now known that to reach the solved position of the rubik cube from any other position you need at most $20$ moves (a $180^{\circ}$ of a face is counted as one move , not two ) . </p>
<p>See for example here : <a href="http://www.cube20.org/" rel="nofollow">http://www.cube20.org/</a> </p>
<blockquote>
<p>Now... | Servaes | 30,382 | <p>Let $G$ the Rubik's cube group of legal moves of the Rubik's cube under concatenation. Then there is a one-to-one correspondence between the set $P$ of legal positions of the Rubik's cube, and the group $G$, by associating to every position $X\in P$ a concatenation of moves $\varphi(X)\in G$ leading from the startin... |
211,052 | <p>I want to modify a list of table row by adding an extra column. For that I <a href="https://reference.wolfram.com/language/ref/Map.html" rel="nofollow noreferrer"><code>Map</code></a> the data with a pure function that evluate the new column value from exisiting one and reconstruct a new list from that and the init... | Sylvain Leroux | 68,791 | <p>While posting my question, the system suggested me the <a href="/questions/tagged/concatenation" class="post-tag" title="show questions tagged 'concatenation'" rel="tag">concatenation</a>, which in its turn lead me to the <a href="https://reference.wolfram.com/language/ref/Catenate.html" rel="nofollow norefe... |
211,052 | <p>I want to modify a list of table row by adding an extra column. For that I <a href="https://reference.wolfram.com/language/ref/Map.html" rel="nofollow noreferrer"><code>Map</code></a> the data with a pure function that evluate the new column value from exisiting one and reconstruct a new list from that and the init... | kglr | 125 | <pre><code>{## & @@ #[[1 ;; 3]], #[[1]] > 2 b, #[[4]]} & /@ data
</code></pre>
<p>or</p>
<pre><code>{Sequence @@ #[[1 ;; 3]], #[[1]] > 2 b, #[[4]]} & /@ data
</code></pre>
<blockquote>
<p>{{1, 45000., 27500., False, "Inverted"},<br>
{2, 22500., 18333.3, False,
""},<br>
{3, 15000., 13750... |
2,846,928 | <blockquote>
<p>If I have the a complex number $z \in \mathbb{C}$ with absolute value $|z| = 1$, how do I show that $-i \frac {z-1}{z+1}$ is real? </p>
</blockquote>
| fleablood | 280,126 | <p>Well $\overline{z + 1} =Re(z+1) - iIm(z+1) = (Re z + 1) - iIm(z) = Re(z) - iIm(z) + 1 = \overline z + 1$ for all complex numbers.</p>
<p>SO</p>
<p>$-i \frac {z-1}{z + 1}= -i\frac {(z-1)(\overline z +1)}{(z+1)\overline{(z+1)}}$</p>
<p>$=-i \frac {z\overline z -\overline z + z - 1}{|z + 1|^2}$</p>
<p>$= -i \frac {... |
3,164,343 | <p>Machine generates one random integer in range <span class="math-container">${[0;40)}$</span> on every spin.<br>
You should choose 5 numbers in that range. </p>
<p>Then the machine will spit out <strong>5</strong> numbers (<em>numbers are independent of each other</em>). </p>
<p><strong>what is the probability t... | Balakrishnan Rajan | 652,935 | <p>Let's say, you have chosen 5 numbers. Then the odds of the machine generating one of your chosen numbers is <span class="math-container">$\frac{5}{40}$</span>. This happens twice and it doesn't happen thrice. So this should be <span class="math-container">$\left(\frac{5}{40}\right)^2\times\left(\frac{35}{40}\right)^... |
3,745,452 | <p>I'm evaluating the area of a parametric function and I'm stuck at this point.
<span class="math-container">$$S=2\pi \int_{0}^{2\pi} x\cos(x)\sqrt{x^2+1} \mathrm{d}x.$$</span>
Is there anyway to work this problem further without a calculator?</p>
| Frosty | 780,483 | <p>This question is not possible to integrate without a calculator. It was in the instructions for the assignment and I overlooked it. Here is the original problem for anyone curious.
<a href="https://www.bartleby.com/solution-answer/chapter-102-problem-57e-calculus-early-transcendentals-8th-edition/9781285741550/set-... |
7,354 | <p>some of my students refer to there being an invisible $-1$ in front of the expression $-(x + 4)$ or in the exponent of $x$. While it is not phrased mathematically, I am ok with them saying this because it reminds them to distribute fully before simplifying etc. It got me thinking though is there a reason to not te... | Karl | 4,668 | <p>I see this as two separate questions:</p>
<p>1) Are there any downsides?</p>
<p>2) Should I condone it for my students?</p>
<p>1) As mentioned in other answers, it can make the transition into deeper areas more difficult than need be. (Understanding for groups)</p>
<p>2) I think the answer to the questions can ... |
7,354 | <p>some of my students refer to there being an invisible $-1$ in front of the expression $-(x + 4)$ or in the exponent of $x$. While it is not phrased mathematically, I am ok with them saying this because it reminds them to distribute fully before simplifying etc. It got me thinking though is there a reason to not te... | Rusty Core | 7,930 | <p>Like another poster, I also see two separate questions here, but different ones.</p>
<p>1) Should we explicitly write <strong>identity value</strong>? 1 is identity for multiplication and division, while 0 is identity for addition and subtraction. Adding 0 does not change the result, likewise dividing by 1 does not... |
2,065,027 | <p>$$\newcommand{\gcd}{\text{gcd}}$$</p>
<blockquote>
<p>Prove: if $d=\gcd(m,n)$ so $\gcd\left(\frac{m}{d},\frac{n}{d}\right)=1$</p>
</blockquote>
<p>Intuitively it is obvious, but I am having a hard time to formalize the proof, what I have came to this:</p>
<p>$d=\gcd(m,n)$ so $d|m$ and $d|n$ therefore $m=dx$ and... | Anurag A | 68,092 | <p>If $d=\gcd(m,n)$, then $\exists \, x,y \in \mathbb{Z}$ such that $d=mx+ny$. From this we can also write
$$1=\frac{m}{d}x+\frac{n}{d}y.$$
Thus $\gcd(m/d,n/d)=1$.</p>
|
1,736,502 | <p>I start with a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$. I assume that the filtration is right-continuous. On this probability space I define a supermartingale $M$. Now suppose that I can find another process $N$ such that $N$ is a cadlag modification of $M$. Hence it is also a supermartin... | John Dawkins | 189,130 | <p>Because $N$ is a modification of $M$, $E(M_t)=E(N_t)$ for all $t\ge 0$. What do you know about the right continuity of $t\mapsto E(N_t)$?</p>
|
153,186 | <p>How can I prove that the double negation elimination is not provable in constructive logic?<br>
To clarify, double negation elimination is the following statement: </p>
<p>$$\neg\neg q \rightarrow q$$</p>
| Zhen Lin | 5,191 | <p>There are two ways to go about this:</p>
<ol>
<li><p><strong>Proof theory.</strong> We analyse the formal proofs in the inference system given, and show that there can be no formal proof of double negation elimination. This is difficult and depends a lot on the details of the inference system.</p></li>
<li><p><stro... |
153,186 | <p>How can I prove that the double negation elimination is not provable in constructive logic?<br>
To clarify, double negation elimination is the following statement: </p>
<p>$$\neg\neg q \rightarrow q$$</p>
| H. Kabayakawa | 32,428 | <p>There is an intuitionistic idea which becomes more valuable with the arrival of computational-mathematics. Roughly speaking in constructive logic or math, when you say "object A" exists you are saying "we know some way to build up A". This position implies the '<em>tertio excluso</em>' rule $\neg(\neg q))\rightarrow... |
130,235 | <p>I need to find </p>
<p>$$f(n) = \int^\infty_0 t^{n-1} e^{-t} dt$$</p>
<p>So I think I find the indefinate integral first? But what do I do with $n$, since I am integrating with respect to $t$?</p>
| kspacja | 22,289 | <p>I think $n$ is parametr. So you have to do it, treating $n$ like normal number.</p>
|
1,815,918 | <p>Let $G$ be a group and $|G|=p^n$ for some prime $p$. If $f:G\to H$ is a surjective homomorphism, how do I know $H=f(G)$ also has cardinality a power of $p$?</p>
| wgrt | 317,545 | <p>I did so</p>
<p>Be $X = P \cap I$ with $P = pair$ and $I= odd$ in [-1,1] </p>
<p>$f \in X, -f \in X$ exist $f(t)$ any $t \in [-1,1]$ such that $f(-t)$ </p>
<p>Thus $f(t)= \dfrac{f(t)+f(-t)}{2}+\dfrac{f(t)-f(-t)}{2}$</p>
<p>Be $ z \in P \cap I$, then $z(t)=z(-t)=-z(-t)$, any $t \in [-1,1] $. </p>
<p>So, $z = 0$... |
3,066,762 | <p>This is a very interesting identity but I don't know how to prove this,
note that <span class="math-container">$$A_1,\ldots,A_J,B \in \mathbb{R}^{n \times n}$$</span>
and <span class="math-container">$m$</span> is the number of block diagonal in <span class="math-container">$\mathbf{A}$</span> ,so consider
<span cl... | Robert Israel | 8,508 | <p>Maple agrees with G&R on the formula, however it has considerable difficulty with the numerical evaluation of the integral as this oscillates very rapidly. But the integral from <span class="math-container">$0$</span> to <span class="math-container">$20$</span> is given as <span class="math-container">$.65366340... |
2,982,205 | <p>I need help with a question that asks this:</p>
<p>Let <span class="math-container">$f(x)= -1/x$</span> and <span class="math-container">$g(x)=e^x$</span> find the domain , range ,monotonicity intervals. Draw the graph of <span class="math-container">$g(f(x))$</span>. </p>
<p>Everything else except the graph is p... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$(h+z)^2 (\bar h +\bar z) - z^2\bar z=(h^2+2hz+z^2)(\bar h +\bar z) - z^2\bar z=$$</span></p>
<p><span class="math-container">$$=h^2\bar h+2h\bar hz+z^2\bar h+h^2\bar z+2hz\bar z+z^2\bar z- z^2\bar z=$$</span></p>
<p><span class="math-container">$$=h^2\bar h+2h\ba... |
3,646,694 | <p>Let <span class="math-container">$M$</span> be an elliptic element of <span class="math-container">$SL_2(\mathbb R)$</span>. Then it is conjugate to a rotation <span class="math-container">$R(\theta)$</span>. Note that we can calculate <span class="math-container">$\theta$</span> in terms of the trace of <span class... | lhf | 589 | <p>Here is another, slightly more sophisticated, take:</p>
<p>The other root is <span class="math-container">$\beta= \gamma ^2+2\gamma-4$</span>, where <span class="math-container">$\gamma = \alpha^2+2\alpha-4$</span>.
Expanding <span class="math-container">$\beta$</span> in terms of <span class="math-container">$\alp... |
1,850,978 | <p>My question concerns the following problem:</p>
<blockquote>
<p>Let $K=\mathbb F_7[T]/(T^3-2)$. Show that $X^3-2$ splits into linear factors in $K[X]$.</p>
</blockquote>
<p>Write $K\simeq \mathbb F_7[\alpha]$ for a root $\alpha\in \overline{\mathbb F_7}$ of $X^3-2$. I found out through experimentation that $2\al... | C. Falcon | 285,416 | <p>Notice that $X^3-2$ does not have any root in $\mathbb{F}_7$. Moreover polynomials of $\mathbb{F}_7[X]$ are invariant under the action of the <a href="https://en.wikipedia.org/wiki/Frobenius_endomorphism" rel="nofollow">Frobenius morphism</a> with respect to $\mathbb{F}_7$. Therefore, $\alpha^7$ and $\alpha^{49}$ ar... |
686,296 | <p>Let $S$ be a set of rational numbers that is closed under addition and multiplication, and having the property that for every rational number $r$ exactly one the following three statements is true: $r\in S$, $-r\in S$, $r=0$.</p>
<p>There are a couple of questions I have to prove but I am having trouble with the fi... | Eka | 130,847 | <p>Suppose $0 \in S$ (our hypothesis).</p>
<p>Then $0 \in S$ and $-0 \in S$.</p>
<p>So, at least two (and not exactly one) of the statements is true.</p>
<p>So, $0 \notin S$.</p>
<p>Because our hypothesis leads to a contradiction, the hypothesis cannot be true.</p>
|
2,459,263 | <p>If $p$ and $q$ are two points on a $n$-dimensional manifold $M$, then their tangent spaces are two $n$-dimensional vector spaces. So algebraically they have the same structure, but they are not the same because $p$ and $q$ are different points. I can understand they are not the same by visualization. How is it state... | anomaly | 156,999 | <p>The general construction you're looking for is that of the tangent bundle (or, more generally, a vector or fiber bundle). The individual tangent spaces are isomorphic; they're vector spaces of the same dimension, and there's a canonical basepoint. The important point, though, is that they vary continuously. More spe... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| Siong Thye Goh | 306,553 | <p>\begin{align}
m(m+1)(m+2)(m+3)&=\left[(m)(m+3) \right] \left[ (m+1)(m+2)\right]\\
&=\left[m^2+3m \right] \left[ m^2+3m+2\right]\\
&=\left[(m^2+3m+1)-1 \right] \left[ (m^2+3m+1)+1\right]\\
&=(m^2+3m+1)^2-1
\end{align}</p>
|
36,083 | <p>I'm trying to find a way to prove this:</p>
<p>EDIT: without using LHopital theorem.
$$\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}.$$
Honestly, I didn't come with any good idea.</p>
<p>We know that $\lim_{x\rightarrow 1}x^{1/m}$ is $1$.</p>
<p>I'd love your help with this.</p>
<p>Thank you.</p>
| Community | -1 | <p>$$\frac{x^{1/m}-1}{x^{1/n}-1} = \frac{e^{\log(x^{1/m})}-1}{e^{\log(x^{1/n})}-1} = \frac{e^{\frac1{m}\log(x)}-1}{e^{\frac1{n}\log(x)}-1} = \frac{e^{\frac1{m}\log(x)}-1}{\log(x)} \frac{\log(x)}{e^{\frac1{n}\log(x)}-1}$$</p>
<p>$$\lim_{x \rightarrow 1} \frac{x^{1/m}-1}{x^{1/n}-1} = \lim_{\log(x) \rightarrow 0} \frac{e... |
468,510 | <p>Given $n^2$, how many fourth powers $(x^4)$ are between 0 and $n^2$? </p>
<p>$n,x\in \mathbb{Z}$</p>
<p>Does this just reduce down to how many squares are below $n$?</p>
| Hagen von Eitzen | 39,174 | <p>There are so many as there are squares between $0$ and $n$, so the answer is $\sqrt n$, with up or downrounding depending on whether you count $0$ and $n$ itself (if it is square)</p>
|
3,490,454 | <p>Definition from Munkres, the textbook we are following: If <span class="math-container">$Y$</span> is a compact Hausdorff space and <span class="math-container">$X$</span> is a proper subspace of <span class="math-container">$Y$</span> whose closure equals <span class="math-container">$Y$</span>, then <span class="m... | celtschk | 34,930 | <p>You are missing that you have to consider the intervals in isolation, not as subsets of <span class="math-container">$\mathbb R$</span>. That means there's no interval <span class="math-container">$[1,2]$</span> sitting in between, and also no intervals on either side. And you may move them around and bend them as m... |
4,338,558 | <p>I am not sure if my proof here is sound, please could I have some opinions on it? If you disagree, I would appreciate some advice on how to fix my proof. Thanks</p>
<p><span class="math-container">$X_1, X_2, ..., X_n$</span> are countably infinite sets.</p>
<p>Let <span class="math-container">$X_1 = \{{x_1}_1, {x_1}... | H A Helfgott | 169,068 | <p>Let me carry out @Ninad_Munshi's suggestion in the first case, so that we can compare it with what I did yesterday.</p>
<p>First of all,
<span class="math-container">$$\iint_U \frac{\log x \log y}{x^\alpha y^\alpha \max(x,y)} dx dy =
2 \frac{\partial}{\partial a} \frac{\partial}{\partial b}
\mathop{\iint_U}_{x>y}... |
4,138,564 | <p>Prove that if <span class="math-container">$q_i$</span>'s are polynomials of degree <span class="math-container">$i$</span>,then <span class="math-container">$q_0,...,q_n$</span> are linearly independent.</p>
<hr />
<p>Assume they are not linearly independent, so there is a scalar <span class="math-container">$\lamb... | Kavi Rama Murthy | 142,385 | <p>Suppose some <span class="math-container">$\lambda_i$</span> is not <span class="math-container">$0$</span>. There is a largest integer <span class="math-container">$m \leq n$</span> such that <span class="math-container">$\lambda_m \neq 0$</span> and <span class="math-container">$\lambda_k=0$</span> for <span class... |
3,676,719 | <p>I've recently started to learn about differential equations and I am having a hard time solving any of them.</p>
<p>I feel like I'm missing some steps.</p>
<p>Therefore, could I please know how to solve:</p>
<p><span class="math-container">$(ye^x +y)dx+ye^{(x+y)}dy=0$</span></p>
<p>So far, I've looked around the... | J. W. Tanner | 615,567 | <p><span class="math-container">$A^2=\pmatrix{a&&b\\c&&d}^2=\pmatrix{a^2+bc&&ab+bd\\ac+dc&&bc+d^2}=\pmatrix{0&0\\0&0}$</span></p>
<p><span class="math-container">$\implies b(a+d)=0$</span> and <span class="math-container">$ c(a+d)=0 \implies a+d=0 $</span> or <span class="math-... |
4,482,637 | <p>Is there are a term for a generalized exponential function? The series expansions of sine and cosine look very similar to the exponential function's series expansion</p>
<p><span class="math-container">\begin{align}
& \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\
& \sum_{n = 0}^{\infty} \frac{(-x)^n}{n!} \\
& ... | Átila Correia | 953,679 | <p>The first integral is wrong because <span class="math-container">$x\in\mathbb{R}$</span> need not be a natural number. So the multiplication cannot be split as you did. Similar reasoning applies to the second case, that is to say, since <span class="math-container">$x$</span> need not be a natural number, the interp... |
4,482,637 | <p>Is there are a term for a generalized exponential function? The series expansions of sine and cosine look very similar to the exponential function's series expansion</p>
<p><span class="math-container">\begin{align}
& \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\
& \sum_{n = 0}^{\infty} \frac{(-x)^n}{n!} \\
& ... | ryang | 21,813 | <p>The above comments and answer point out that you can't take a variable out of an integral, and that <span class="math-container">$x$</span> is not a natural number, but in fact, in the given examples, the error could be more fundamental.</p>
<p>Let <span class="math-container">$f:\mathbb Z \to \mathbb R$</span> and ... |
372,420 | <p>Let <span class="math-container">$X$</span> be a finite ultrametric space and <span class="math-container">$P(X)$</span> be the space of probability measures on <span class="math-container">$X$</span> endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula
<span class="... | oliversm | 112,077 | <h1>Star discrepancy</h1>
<p>The <a href="https://mathworld.wolfram.com/StarDiscrepancy.html" rel="nofollow noreferrer">star discrepancy</a> is usually used when thinking about random numbers and low discrepancy sequences, and seems to fit the bill for your task.</p>
|
345,652 | <p>I try to understand why by definition </p>
<ol>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also </li>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .</li>
</ol>
<p>Those are continued fractions, and $1$ and $2$ are notes I have in the lecture summery.</p>
<p>But we can add bracke... | Clive Newstead | 19,542 | <p>It's a fairly easy inductive proof that you can write
$$[c_0, c_1, c_2, \dots, c_{n-1}, c_n] = [c_0, [c_1, [c_2, \cdots [c_{n-1}, c_n] \cdots ]]]$$
and you can add or remove any of the sets of square brackets that appear on the right-hand side as you please. That is, you can stick a $[$ where you like as long as the... |
17,103 | <p>I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point: </p>
<blockquote>
<p>If <span class="math-container">$f$</span> is a quadratic form in <span class="math-container">$X_1,X_2,\cdots,X_n$</span> over a
finite field, then one can change variables such that <... | damiano | 4,344 | <p>I will sketch below a standard argument to show what you need, because I find it very neat!</p>
<p>Let $V$ be a finite dimensional vector space over a field $k$ and let $q \colon V \to k$ be a quadratic form on $V$. Denote by $b$ the symmetric bilinear form associated to $q$: thus for vectors $v,w \in V$ define $b... |
2,552,720 | <p>How would you prove algebraically that the assertion $\exists (x,y) \in (0,1)^2. x +y > 1 + xy$ <a href="http://www.wolframalpha.com/input/?i=x%20%2B%20y%20%3E%20%201%20%2B%20xy,%200%20%3C%20x%20%3C%201,%200%20%3C%20y%20%3C%201" rel="nofollow noreferrer">is false</a>?</p>
| stressed out | 436,477 | <p>Use the factorization
$$1-x-y+xy=(1-x)(1-y)$$</p>
<p>Now note that:</p>
<p>$0<x<1 \implies 0<1-x<1$</p>
<p>$0<y<1 \implies 0<1-y<1$</p>
<p>Since both are positive,</p>
<p>$$1-x-y+xy>0$$
$$1+xy>x+y$$</p>
|
2,552,720 | <p>How would you prove algebraically that the assertion $\exists (x,y) \in (0,1)^2. x +y > 1 + xy$ <a href="http://www.wolframalpha.com/input/?i=x%20%2B%20y%20%3E%20%201%20%2B%20xy,%200%20%3C%20x%20%3C%201,%200%20%3C%20y%20%3C%201" rel="nofollow noreferrer">is false</a>?</p>
| zipirovich | 127,842 | <p>Hint: move all terms to the same side to get the equivalent inequality $1-x-y+xy<0$, and then factor the expression on the left-hand side.</p>
|
15,994 | <p>I am trying to solve this problem <a href="https://mathematica.stackexchange.com/questions/15922/help-me-to-solve-a-condition-from-an-equation-that-requires-many-reductions">here</a> but have to understand basics. Suppose I have a multiplication such as <span class="math-container">$a*(b+c+d+...+x)$</span>. How can ... | rcollyer | 52 | <p>While covered by the comments and <a href="https://mathematica.stackexchange.com/a/16008/52">another answer</a>, I would like to expand on this a bit. The word "simplify" has very different connotations between what you intend and what <em>Mathematica</em> intends. For <em>Mathematica</em>, it means simply the leas... |
2,738,138 | <p>I know it's probably a stupid question, but I'm confused. I have a set {$x\in\mathbb R, \frac{1}{x} \le 1$} that I want to represent as interval/s.</p>
<p>Thinking about it logically, I know that the set is $x\in]-\infty, 0[$U$[1, +\infty[$.</p>
<p>However, when trying to solve the inequality, I can't seem to get ... | user | 505,767 | <p>It is correct indeed the solutions are $x<0$ and $x\ge 1$ since</p>
<ol>
<li>for $x > 0$ we have $x \ge 1\implies x\ge1$</li>
<li>for $x < 0$ we have $x \le 1\implies x<0$</li>
</ol>
|
2,738,138 | <p>I know it's probably a stupid question, but I'm confused. I have a set {$x\in\mathbb R, \frac{1}{x} \le 1$} that I want to represent as interval/s.</p>
<p>Thinking about it logically, I know that the set is $x\in]-\infty, 0[$U$[1, +\infty[$.</p>
<p>However, when trying to solve the inequality, I can't seem to get ... | Mohammad Riazi-Kermani | 514,496 | <p>For positive values of $x$, $$1/x \le 1 \iff x\ge 1$$</p>
<p>For negative values of $x$, $1/x$ is negative so it is less than $1.$</p>
<p>For $x=0$, $1/x$ is undefined </p>
<p>Thus the answer is $$x\in [1,\infty)$$</p>
|
13,861 | <p>I am currently working with a weighted adjacency matrix for a directed graph, and it contains several 0 columns and rows. With the unaltered matrix, I am able to monitor the relations between vertices with,</p>
<pre><code>TableForm[Normal @ WeightedAdjacencyMatrix[graph],
TableHeadings -> {a = VertexList[gr... | kglr | 125 | <pre><code>nonzerorowsF = Function[{grph}, Pick[Range[VertexCount[grph]],
Tr@Abs[#] != 0 & /@ WeightedAdjacencyMatrix[grph]]];
nonzerocolsF = Function[{grph}, Pick[Range[VertexCount[grph]],
Tr@Abs[#] != 0 & /@ Transpose[WeightedAdjacencyMatrix[grph]]]];
</code></pre>
<p>example:</p>
<pre><code> options ... |
4,517,597 | <p>Suppose <span class="math-container">$\{f_n\}_{n \in \mathbb{N}}$</span> is a family of bounded, differentiable, monotone increasing functions on <span class="math-container">$[0,1]$</span>, which converge uniformly to a limit <span class="math-container">$f$</span>. Also, suppose we know that <span class="math-con... | Yuval Peres | 360,408 | <p>The answer is positive, and weaker assumptions suffice.</p>
<p><strong>Claim:</strong> Suppose <span class="math-container">$\{f_n\}_{n \in \mathbb{N}}$</span> is a family of bounded, differentiable, functions on <span class="math-container">$[0,1]$</span>, which converge pointwise to a limit <span class="math-con... |
2,406,049 | <p>I would appreciate help understanding what $\aleph_1$ is according to this definition:</p>
<blockquote>
<blockquote>
<p>If $\alpha$ is an ordinal, then $\aleph_{\alpha}$ is the unique infinite cardinal such that:
$\{\kappa:\kappa\text{ is an infinite cardinal and }\kappa\lt\aleph_{\alpha}\}$
is isomor... | Angina Seng | 436,618 | <p>As $\aleph_1$ is the second infinite cardinal, the infinite cardinals
$\le\aleph_1$ are $\aleph_0$ and $\aleph_1$. It looks like your definition
should have $<$ rather than $\le$.</p>
|
3,792,983 | <p>As far as I can tell the definition of a source and a sink respectively are given in terms of the divergence operator.</p>
<p>That is, given a vector field <span class="math-container">$\vec{D}$</span>, it has a <em>source</em> in point <span class="math-container">$P$</span> if its divergence <span class="math-cont... | Daniel Fischer | 83,702 | <p><span class="math-container">$A$</span> must be assumed nonempty of course. Then we can use induction on the dimension.</p>
<p>In <span class="math-container">$\mathbb{R}^1$</span>, a nonempty closed convex set <span class="math-container">$A$</span> that contains no line has one of the forms <span class="math-conta... |
1,393,955 | <p>I have the following evolution equations realted to mean curavture flow, with the induced metric $g=\{g_{ij}\}$, measure $d\mu$ and second fundamental form $A=\{h_{ij}\}$:</p>
<p>1)$\frac{\partial}{\partial t}g_{ij}=-2Hh_{ij}$</p>
<p>2)$\frac{\partial}{\partial t}d\mu=-H^2d\mu$</p>
<p>3)$\frac{\partial}{\partial ... | Anthony Carapetis | 28,513 | <p>I'm guessing the equality you need is</p>
<p>$$h^i_j \frac{\partial P}{\partial h^i_j} = \alpha P.$$</p>
<p>To see that this is true, note that the LHS is the derivative of $P$ in the outwards radial direction; i.e.</p>
<p>$$ h^i_j \frac{\partial P}{\partial h^i_j} = \frac{d}{dt}\Big|_{t=0} P((1+t)W).$$</p>
<p>I... |
4,168,198 | <p>If M is a connected n-manifold and N is a codim 2 submanifold
then M-N is connected.</p>
<p>Is this true?</p>
<p>I want to show <span class="math-container">$H_0(M-N)=\mathbb Z$</span>. I think it's better to do in <span class="math-container">$\mathbb Z_2$</span> coefficient because we get orientability. But becaus... | Lee Mosher | 26,501 | <p>This has little to do with orientability, nor compactness, Poincare duality, nor <span class="math-container">$\mathbb Z_2$</span> coefficients. It's just proving path connectivity of <span class="math-container">$M-N$</span>. Also, the proof works in any codimension <span class="math-container">$q \ge 2$</span>, an... |
1,811,567 | <p>We had a high school mathematics teacher who taught us a cool technique that I've forgotten. It can be used, for example, for developing a formula for the sum of squares for the first "n" integers. You start by making a column for Sn, and then determine the differences until you get a constant. See the picture.</p>
... | Abr001am | 223,829 | <ul>
<li>backgrounded on the fact that the difference of two consecutive squares is an odd number:</li>
</ul>
<p>$(n+1)^2-n^2=2n+1$</p>
<p>the difference of two consecutive odd numbers is always $2$ thus the self-saying technique.</p>
<p>So when we proceed to find a square $n^2$ we sum up all odd numbers from 1 to $... |
2,936,356 | <p>Let <span class="math-container">$a,b>0$</span>. How can I calculate the roots of the following polynomial?</p>
<p><span class="math-container">$$2bx^6 + 4bx^5+(4b-a)x^4 + 4(b+a)x^3 + (2b-6a)x^2+4ax-a=0$$</span></p>
| Robert Israel | 8,508 | <p>There is really only one parameter here rather than two: if <span class="math-container">$s = a/b$</span>, you can divide by <span class="math-container">$b$</span> and write the equation as
<span class="math-container">$$ 2\,{x}^{6}+4\,{x}^{5}- \left( s-4 \right) {x}^{4}+4\, \left( s+1
\right) {x}^{3}-2\, \left( ... |
2,694,122 | <blockquote>
<p>Given that a function $f$ is defined as $$f(x)=1+2x+3x^2+4x^3+...$$. We have to prove that $f$ is continuous on $[0,\frac{1}{8}]$ and evaluate $$\int_{0}^{\frac{1}{8}}f(x)dx$$ </p>
</blockquote>
<p>I am not sure in a particular area. I have proved $f$ is continuous on $[0,\frac{1}{8}]$. But the evalu... | user284331 | 284,331 | <p>$f(x)=\displaystyle\sum_{n=0}^{\infty}(n+1)x^{n}$, $x\in[0,1/8]$, then $f(x)\leq\displaystyle\sum_{n=0}^{\infty}(n+1)/8^{n}<\infty$ by ratio test, and by Weierstrass M-Test, the series is uniformly convergent, and we are allowed to swipe the sum and integral: $\displaystyle\int_{0}^{1/8}f(x)dx=\displaystyle\sum_{... |
2,694,122 | <blockquote>
<p>Given that a function $f$ is defined as $$f(x)=1+2x+3x^2+4x^3+...$$. We have to prove that $f$ is continuous on $[0,\frac{1}{8}]$ and evaluate $$\int_{0}^{\frac{1}{8}}f(x)dx$$ </p>
</blockquote>
<p>I am not sure in a particular area. I have proved $f$ is continuous on $[0,\frac{1}{8}]$. But the evalu... | Jack D'Aurizio | 44,121 | <p>As an alternative approach, it is not difficult to notice that $(1-x)^2 f(x)$ has a simple expression.<br>
In general, if $f(x)=\sum_{n\geq 0}p(n) x^n$ with $p$ being a polynomial with degree $d$, $(1-x)^{d+1}f(x)$ has a simple expression, since the multiplication by $(1-x)$ acts on the coefficients of $f(x)$ as the... |
2,268,225 | <p>I want to investigate the convergence behavior of $$\int_{0}^{\infty} \cos(x^r)\, dx \hspace{5mm} \textrm{and} \hspace{5mm} \int_{\pi}^{\infty} \left(\frac{\cos x}{\log x}\right)\arctan\lfloor x\rfloor dx.$$ My theoretical tools are Abel's test and Dirichlet's test:
Say I have an integral of the form $$\int_{a}^{b... | k.Vijay | 428,609 | <p>Apply $(1-x)^{-2}=1+2x+3x^2+4x^3+\cdots\infty$.</p>
<p>Now, </p>
<p>$\dfrac{1}{4}+\dfrac{2}{8}+\dfrac{3}{16}+\dfrac{4}{32}+\dfrac{5}{64}+\cdots\infty\\
=\dfrac{1}{4}\left(1+\dfrac{2}{2}+\dfrac{3}{2^2}+\dfrac{4}{2^3}+\dfrac{5}{2^4}+\cdots\infty\right)\\
=\dfrac{1}{4}\left(1-\dfrac{1}{2}\right)^{-2}\hspace{25pt}\tex... |
54,088 | <p>I am studying KS (Kolmogorov-Sinai) entropy of order <em>q</em> and it can be defined as</p>
<p>$$
h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right)
$$</p>
<p>Why is it defined as supremum over all possible partitions <em>P</em> and not maximum? </p>
<p>When do people use supremum and w... | Eric Naslund | 6,075 | <p>If the maximum exists, then the supremum and maximum are the same. However sometimes the maximum does not exist, and there is no maximal element. In this case it still makes sense to talk about a least upper bound.</p>
<p>The classic example is the set of all rationals whose square is less than or equal to $2$. ... |
211,297 | <p>Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties:</p>
<ol>
<li>For each $ x \in \mathbb{F} $, the function $ f(x,\cdot): y \mapsto f(x,y) $ is a polynomial function on $ \mathbb{F} $.</li>
<li>For each $ y \in \mathbb{F} $, ... | Gerry Myerson | 8,269 | <p>Maybe this works for the countably infinite case. Order the rationals (or whatever countably infinite field you have) as $r_1,r_2,\dots$. Let $$f(x,y)=(x-r_1)(y-r_1)+(x-r_1)(x-r_2)(y-r_1)(y-r_2)+\cdots$$ Then if $r$ is any rational, say, $r=r_j$, then $f(r,y)$ is a polynomial of degree $j-1$ in $y$, and similarly fo... |
2,497,072 | <p>Write using logical symbols: "Every even number greater than two is the sum of two prime numbers" <br>
This is how I attempted to write this:
$$((n=2m \land n > 2) \Rightarrow n=p_1+p_2)\land(x|p_1 \Rightarrow x=1 \lor x=p_1) \land (y|p_2 \Rightarrow y=1 \lor y=p_2) \land (p_1 >1 \land p_2 >1)$$
I am not su... | fonini | 113,664 | <p>(I'm assuming this is homework; by the way, you should provide more context in your question, like where the question came from, and any restrictions like "quantifiers are not allowed".)</p>
<p>Anyway, I don't know what your teacher wants you to do, but I think you should be aware that you <em>can't</em> write "Eve... |
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | gen-ℤ ready to perish | 347,062 | <p>Perhaps you create a mapping from the length of each segment to the number of segments necessary, namely $n= (1 \ \mathrm{cm})/{\ell}$. Graph this function with $n$ on the vertical axis and $\ell / \mathrm{cm}$ on the horizontal axis. Show them that where $\ell = 0 \ \mathrm{cm}$, which is the “length” of a point, i... |
2,417,029 | <p>I have been told that a line segment is a set of points.
How can even infinitely many point, each of length zero, can make a line of positive length?</p>
<p>Edit:
As an undergraduate I assumed it was due to having uncountably many points.
But the Cantor set has uncountably many elements and it has measure $0$.</p>
... | Cort Ammon | 153,899 | <p>First off, consider instead trying to argue that a line can be made up of infinitely many small lines, each of infintessimally short length. This will be closer to the way calculus handles lines, so will serve them well in the future.</p>
<p>If you must use points, one approach you might be able to take is to star... |
657,992 | <p>I understand how to explain but can't put it down on paper.</p>
<p>$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.</p>
<p>I let $\displaystyle x = \frac{a}{b}$ and $\displaystyl... | Joe Anderson | 249,953 | <p>I wanted to add to Cameran Buie's answer with the actual proof by induction, since it seems that the question was "How do I word the proof?" I'm going overboard here, being fully precise; in any normal situation, much of this reasoning can be brushed over and still get the point across.</p>
<h3>First, a precise wor... |
2,938,491 | <p>I have been pondering this question for a little while, and unfortunately the Google has not given me an answer.</p>
<p>I understand that for example you had a table or graph that crosses or touches the x axis at say x= -2, 0, 3 you could form an equation as</p>
<p>f(x) = ax(x+2)(x-3)</p>
<p>Then solve for a and ... | Andrei | 331,661 | <p><span class="math-container">$$S(n)=(1+2n)^2=1+4n+4n^2$$</span>
You can now use the following <span class="math-container">$$\sum_{n=0}^m1=m+1\\\sum_{n=0}^mn=\frac{m(m+1)}{2}\\\sum_{n=0}^mn^2=\frac{m(m+1)(2m+1)}{6}$$</span></p>
<p>Alternatively, compute the first 4-5 elements. The sum of a polynomial of order <span... |
148,612 | <p>You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?</p>
<p>This is my first question on this site, so if ... | TonyK | 1,508 | <p>$\hskip 2in$ <img src="https://i.stack.imgur.com/EjFV4.jpg" alt="enter image description here"></p>
<p>We are given that $ZY = BC$, and we want to find $BC/AB$. To keep things simple, choose units so that $AB=1$; and let $s = AC$ and $t = BC$. Then we want to find $t$.</p>
<p>Triangle $XYC$ is similar to triangle ... |
162,364 | <p>Assuming we have a list below, it has real number elements and complex numbers, how can I quickly find if the list has any Real number that its value is less than 1.0? </p>
<pre><code> lis = {3 Sqrt[354], Sqrt[2962], Sqrt[2746], 3 Sqrt[282], Sqrt[2338],
Sqrt[2146], 3 Sqrt[218], Sqrt[1786], Sqrt[1618], 27 Sqrt[2]... | aardvark2012 | 45,411 | <p>There's always <code>Select</code>:</p>
<pre><code>Select[lis, Negative[# - 1] &]
</code></pre>
<p><a href="http://reference.wolfram.com/language/ref/Negative.html" rel="nofollow noreferrer"><code>Negative</code></a> is nice because it'll work on pretty much anything, but only return <code>True</code> if the a... |
4,292,822 | <blockquote>
<p>Prove linear independence of <span class="math-container">$1+x^3-x^5,1-x^3,1+x^5$</span> in the Vector Space of Polynomials</p>
</blockquote>
<p>The attempts I found online all are quite easy. You just substitute something in for <span class="math-container">$x$</span> into the equation <span class="mat... | Sebastián P. Pincheira | 721,724 | <p>Let's play a little with notation: let <span class="math-container">$$\begin{align}&p_1\colon x\mapsto 1+x^3-x^5,\\&p_2\colon x\mapsto 1-x^3\text{ and}\\&p_3\colon x\mapsto 1+x^5.\end{align}$$</span></p>
<p>What you want to show is that, for <span class="math-container">$a,b,c\in\mathbb{R}$</span>
<span ... |
68,147 | <p>While I was investigating some specific types of prime numbers I have faced with the following infinite sequence :</p>
<p>$1,2,8,9,15,20,26,38,45,65,112,244,303,393,560,....$</p>
<p>I tried to find recursive formula using Maple and it's listtorec command, so up to $393$ I got the next output:</p>
<p>$ f(n+3) = ((... | spin | 12,623 | <p>You can look up integer sequences at OEIS: <a href="http://oeis.org/A056805">http://oeis.org/A056805</a></p>
<p>So your sequence is "Numbers $n$ such that $6*10^n+1$ is prime". I assume you're looking for a formula, but if there was a closed-form expression for these numbers, we could find arbitrarily large prime n... |
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | András Bátkai | 61 | <p>I think if you notice this problem, you should briefly revise it, go on with the material, stress that it is essential that they know the basic stuff, and hand out extra exercises on the basic material which will not be graded but which you are ready to discuss in your office hour.</p>
<p>This way you offer a chanc... |
1,867,695 | <blockquote>
<p>$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$</p>
</blockquote>
<p>$\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \right ) = \infty * 0 = 0$</p>
<p>Did I do it correctly?</p>
<p>Problem is when ... | egreg | 62,967 | <p>No, you can't do it like that: when you have a product where one factor has limit $\infty$ and the other has limit $0$, you cannot apply a theorem on products of limits.</p>
<p>The theorem helps when both limits are finite and you just multiply them; it also works when one limit is $\infty$ (or $-\infty$) and the o... |
183,237 | <blockquote>
<p>Does</p>
<p><span class="math-container">$$\int_{-\infty}^\infty \text{e}^{\ a\ (x+b)^2}\ \text dx=\int_{-\infty}^\infty \text{e}^{\ a\ x^2}\ \text dx\ \ \ \ \ ?$$</span></p>
<p>hold, even if the imaginary part of <span class="math-container">$b$</span> is nonzero?</p>
</blockquote>
<p>What I really wan... | paul garrett | 12,291 | <p>Another approach that sometimes simplifies this sort of issue is to invoke the "identity principle": when the integral is holomorphic as a function of the parameter (some potential danger here!), the outcome can be computed in a convenient range, and then invoke the identity principle to know that the same formula h... |
2,787,114 | <p>Before going to my question, let me give two prelimilary definitions</p>
<blockquote>
<p><strong>Definition 1.</strong> Let $S\subseteq\mathbb{R}^n$ be a non=empty open set in $\mathbb{R}^n$ under the usual topology on $\mathbb{R}^n$ and $f:S\to \mathbb{R}^m$. Let $\mathbf{c}\in S$ and $g:U(\subseteq \mathbb{R})\... | Eric Wofsey | 86,856 | <p>If $f$ has an approach independent derivative at $c$, then $f$ is constant in a neighborhood of $c$ (and therefore rather trivially is continuous at $c$).</p>
<p>Indeed, suppose $f$ is not constant in any neighborhood of $c$. That means there is a sequence $(c_n)$ approaching $c$ such that $f(c_n)\neq f(c)$ for al... |
3,687,355 | <p>I have trouble with how to find eigenvectors when you have a complex eigenvalue</p>
<p>For example the matrix
<span class="math-container">$$ \begin{pmatrix}
0 & 1\\
-2 & -2
\end{pmatrix}$$</span></p>
<p>Here you get the eigenvalues <span class="math-container">$-1$</span> and <span class="math-container">... | José Carlos Santos | 446,262 | <p>You do it just as you would for real eigenvalues.</p>
<p>For instance, if you want an eigenvector corresponding to the eigen value <span class="math-container">$-1+i$</span>, you solve the system<span class="math-container">$$\left\{\begin{array}{ll}y=(-1+i)x\\-2x-2y=(-1+i)y,\end{array}\right.$$</span>you will get ... |
75,704 | <p><strong>Bug introduced in 10.0 and fixed in 10.0.2</strong></p>
<hr>
<p>I was looking at integrals like:</p>
<pre><code>Integrate[HermiteH[50, x]*Exp[-x^2], {x, 0, Infinity}]
</code></pre>
<p>which gave me a "does not converge on $(0,\infty)$" error. On the other hand something like </p>
<pre><code>Integrate[(x... | Silvia | 17 | <p>You can use the new <code>RunProcess</code> as you have already noticed:</p>
<pre><code>RunProcess[{"powershell", "Get-Help"}, "StandardOutput"]
</code></pre>
<p>Or you can stick to the old fashion (which is basically invoke powershell from cmd):</p>
<pre><code>RunInCmd[cmd_] :=
FromCharacterCode[ReadLis... |
75,704 | <p><strong>Bug introduced in 10.0 and fixed in 10.0.2</strong></p>
<hr>
<p>I was looking at integrals like:</p>
<pre><code>Integrate[HermiteH[50, x]*Exp[-x^2], {x, 0, Infinity}]
</code></pre>
<p>which gave me a "does not converge on $(0,\infty)$" error. On the other hand something like </p>
<pre><code>Integrate[(x... | Gustavo Delfino | 251 | <p>This is another way of doing it:</p>
<pre><code>Import["!powershell.exe \"Get-Help\"", "TEXT"]
</code></pre>
|
872,893 | <p>How can I prove that the equation $r=16^r$ is wrong for any arbitrary value of $r$?
I have tried:</p>
<p>\begin{align}
&r=16^r &&\implies \log_r r = \log_r 16^r \\
&&& \implies 1 = r\log_r 16 \\
&&& \implies 1/r = \log_r 16 \\
&&& \implies r^{1/r} = r^{\log_r 16}\\
&a... | user121591 | 121,591 | <p>Draw a graph of y=x and y=16^x on the same axes, then I'm sure you can find a reason why there can't be a solution</p>
|
1,314,802 | <p>True.</p>
<p>Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume:</p>
<p>$$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$</p>
<p>Since $ \lim_{n\to\infty} \frac{1}{n} $ is bounded in $ (0,1] $ and $ I \subset (0,1] $, we have by hypothesis $f$ uniformly continuous the... | marwalix | 441 | <p><strong>Hint</strong> look at the $\log$ then use $\cos{x}=1-\frac{x^2}{2}+o(x^3)$ and $\log(1+x)=x-\frac{x^2}{2}+o(x^2)$</p>
<p>Let's look in more details</p>
<p>$$\log\left(\left(1+\frac{1-\cos x}{x}\right)^{\frac{1}{x}}\right)=\frac{1}{x}\log\left(1+\frac{1-\cos x}{x}\right)$$</p>
<p>Using the expansion of $\c... |
1,314,802 | <p>True.</p>
<p>Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume:</p>
<p>$$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$</p>
<p>Since $ \lim_{n\to\infty} \frac{1}{n} $ is bounded in $ (0,1] $ and $ I \subset (0,1] $, we have by hypothesis $f$ uniformly continuous the... | xpaul | 66,420 | <p>You can use this way:
\begin{eqnarray}
\lim\limits_{x\to 0} \left(1+\frac{1-\cos x}{x}\right)^{\frac{1}{x}}&=&\lim\limits_{x\to 0} \left(\left(1+\frac{1-\cos x}{x}\right)^{\frac{x}{1-\cos x}}\right)^{\frac{1-\cos x}{x^2}}=e^{\frac12}
\end{eqnarray}
This is because
$$ \lim_{x\to0}\frac{1-\cos x}{x}=0, \lim_{... |
2,393,130 | <blockquote>
<p><strong>Problem:</strong> <span class="math-container">$ABCD$</span> is a rectangle. A point <span class="math-container">$P$</span> is <span class="math-container">$11$</span> from <span class="math-container">$A$</span>, <span class="math-container">$13$</span> from <span class="math-container">$B$</s... | farruhota | 425,072 | <p>Hint: draw lines through <span class="math-container">$P$</span> parallel to the sides of rectangle. Use Pythagoras theorem for the lines <span class="math-container">$PA, PB, PC, PD$</span> to make up the system of four equations. Then you get the answer <span class="math-container">$1$</span>.</p>
<p>Details:</p>
... |
2,744,008 | <p>Since the isolated singularities are $z=k\pi, k \in \mathbb Z$, so we divided the complex plane into the disjoint annulus, i.e. $\{z: n\pi <|z|<(n+1)\pi\}, n \in \mathbb N \cup \{0\}$. On these annuli, $f$ is analytic, so has a unique Laurent series.</p>
<p>First, let's consider $\{z: 0 < |z|<\pi\}$, th... | Andrew Gross | 370,491 | <p>By $f(x)$ you're referring to the probability density function (PDF), which tells you the probability of observing a single value. </p>
<p>The expected value, call it $\mathbb{E}[X]$, is the long-run average. For a continuous uniform distribution running from 0 to 1 this is defined as:
$$\mathbb{E}[X]=\int_{0}^{1}x... |
1,840,176 | <p>I got a task in front of me but I don't really understand it. If someone could explain, I think I would be able to solve it myself.</p>
<blockquote>
<p>$P(x) = \sum_{k=0}^{\infty}a_{k}x^{k}$ is a power series. There exists a $k_{0} \in \mathbb{N}$ with $a_{k} \neq 0$ for all $k \geq k_{0}$.</p>
<p>Proof that... | user21820 | 21,820 | <p><strong>Hint</strong>: Since you've gotten explanation of the notation already, here's the sketch of how to prove the result. Radius of convergence means that the power series converges for anything <strong>strictly</strong> inside. So if $|x| = r < \frac1a$, then let $s$ be such that $r < s < \frac1a$, and... |
1,789,051 | <p>I know how to prove when the two equivalence classes are not disjoint, i.e. $[a]=[b]$. I see that the proof works for proving a equivalence class is disjoint, but I don't get it. Can someone explain it to me? </p>
| ervx | 325,617 | <p>Suppose we have two equivalence classes $[a]\not=[b]$. We'll show they are disjoint. Suppose $x\in[a]\cap[b]$. Then, $x\sim a$ and $x\sim b$. By symmetry, $a\sim x$, and hence, by transitivity, $a\sim b$. Therefore, $[a]=[b]$, which is a contradiction. </p>
|
1,789,051 | <p>I know how to prove when the two equivalence classes are not disjoint, i.e. $[a]=[b]$. I see that the proof works for proving a equivalence class is disjoint, but I don't get it. Can someone explain it to me? </p>
| student forever | 328,820 | <p>Thinking as equivalence relation on $\mod n$:</p>
<p>If we have, for fixed integers $k,l$, $$a+kn=b+ln$$ holds then for arbitrary integer $t$ we have $$b+tn=a+(k-l+t)n$$ and hence $$[b] \subseteq [a] \text{ and then similarly } [a] \subseteq [b] .$$</p>
<p>General is similar.</p>
|
120,281 | <p>Does this outline of a proof work?</p>
<p>Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to show that they are not isometric. </p>
<p>One way to distinguish the two spaces is their sectional curvature.... | Claudio Gorodski | 15,155 | <p>In general, the ball $B^n$ with the Bergmann metric is isometric to the Hermitian symmetric space $SU(1,n)/S(U(1)\times U(n))$ where $SU(1,n)$ denotes the special pseudo-unitary group
with respect to the indefinite Hermitian inner product with signature $(1,n)$ and $S(U(1)\times SU(n))$ denotes the subgroup of bloc... |
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