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 |
|---|---|---|---|---|
3,934,974 | <p>I'm wondering how to prove the associativity and identity to prove that the Möbius transformations forms a group.</p>
<p>A Möbius transformation is a complex function of the form <span class="math-container">$M(z)=\dfrac{az+b}{cz+d}$</span>.</p>
<p>Thank you in advance :)</p>
| John Clever | 783,246 | <p>I don't know how much you've done on your own, but to help subsequent users, I'll put the full answer. Stop after each hint and try and do it yourself.</p>
<p><strong>Hint 1:</strong>
Prove that the Möbius transformations are bijective from the extended complex numbers to themselves.</p>
<p><strong>Hint 2:</strong>
... |
1,176,098 | <p>Here are some of my ideas:</p>
<p><strong>1. Addition Formula:</strong> <span class="math-container">$\sin{x}$</span> and <span class="math-container">$\cos{x}$</span> are the unique functions satisfying:</p>
<ul>
<li><p><span class="math-container">$\sin(x + y) = \sin x \cos y + \cos x \sin y $</span></p>
</li>
<li... | Emilio Novati | 187,568 | <p>My preferred definition is:</p>
<blockquote>
<p>$\cos x$ and $\sin x$ are the real and imaginary parts of the
exponential function $\exp(ix)$.</p>
</blockquote>
<p>Since we have:
$$
\begin{split}
e^{ix}= \sum_{k=0}^\infty\dfrac{(ix)^k}{k!}&=1+ix+\dfrac{(ix)^2}{2!}+\dfrac{(ix)^3}{3!}+\cdots+\dfrac{(ix)^n}... |
1,176,098 | <p>Here are some of my ideas:</p>
<p><strong>1. Addition Formula:</strong> <span class="math-container">$\sin{x}$</span> and <span class="math-container">$\cos{x}$</span> are the unique functions satisfying:</p>
<ul>
<li><p><span class="math-container">$\sin(x + y) = \sin x \cos y + \cos x \sin y $</span></p>
</li>
<li... | Math2718 | 685,688 | <p>One definition that I think has a very clear geometric meaning is this: On the unit circle in the <span class="math-container">$xy$</span>-plane, draw a line segment from the origin to a point on the circle. Call the angle that the line segment makes with the <span class="math-container">$x$</span>-axis <span class=... |
1,461,254 | <p>I am about to take an undergraduate course in Mathematical logic any textbooks to recommend.Want it to be rigorous and not missing things. I am an Math undergraduate. Got still 2 years for my degree. I have taken courses mostly in Algebra (ring theory etc.). Also taken Real analysis courses. So my mathematical matur... | David | 297,532 | <p>I would recommend Cori and Lascar's <a href="http://rads.stackoverflow.com/amzn/click/0198500483" rel="nofollow"><em>Mathematical Logic: A Course with Exercises</em>.</a> It has a lot of examples from algebra, so is appropriate for a math major who has taken abstract algebra.</p>
|
16,733 | <p>This is a variant on the question <a href="https://matheducators.stackexchange.com/questions/14492/small-real-numbers">small real numbers</a>.</p>
<p>I have a disagreement with someone about the meaning of "bigger" real numbers.</p>
<p>Say we have the real number <span class="math-container">$-1.$</span> Is <span ... | amWhy | 238 | <p>The words "big" and "small" are relative. Asking a student: "Is <span class="math-container">$1$</span> small?" is confusing, and meaningless. Likewise, "bigger" and "smaller" are relative. If you consider six numbers, <span class="math-container">$1, 2, 4, 5, 7, 8$</span>, then each of <span class="math-container... |
2,091,589 | <p>The remainder when a polynomial $f(x)$ is divided by $(x-2)(x+3)$ is $ax+b$. When $f(x)$ is divided by $(x-2)$, then remainder is $5$. $(x+3)$ is a factor of $f(x)$. Find the values of $a$ and $b$. I am thinking of using the remainder and factor theorem to solve this however their quotients are different. Can anyone... | Ian S | 405,681 | <p>Yes you are on the right track.</p>
<p>Let <em>f</em>(<em>x</em>) = (<em>x</em>-2)(<em>x</em>+3) x <em>m</em> + (<em>ax</em> + <em>b</em>).</p>
<p>We know <em>f</em>(2) = 5, so (2-2)(2+3) x <em>m</em> + (2 <em>a</em> + <em>b</em>) = 5
And similarly <em>f</em>(-3) = 0, so (-3-2)(-3+3) x <em>m</em> + (-3 <em>a</em> ... |
2,091,589 | <p>The remainder when a polynomial $f(x)$ is divided by $(x-2)(x+3)$ is $ax+b$. When $f(x)$ is divided by $(x-2)$, then remainder is $5$. $(x+3)$ is a factor of $f(x)$. Find the values of $a$ and $b$. I am thinking of using the remainder and factor theorem to solve this however their quotients are different. Can anyone... | Roman83 | 309,360 | <h3>Hint:</h3>
<p>If <span class="math-container">$$f(x)=(x-\alpha)g(x)+r$$</span>
then <span class="math-container">$r=f(\alpha)$</span></p>
<p>Really, <span class="math-container">$x=\alpha$</span> <span class="math-container">$f(\alpha)=0+r$</span></p>
<p>Then <span class="math-container">$$f(x)=(x-2)(x+3)+ax+b$$</s... |
1,509,007 | <blockquote>
<p>Consider an unweighted and undirected graph $G=(V,E)$, where the vertices $V$ of $G$ lie on the unit n-sphere. If we choose a normal vector uniformly at random on this $n$-sphere, then the corresponding hyperplane (shifted to go through the origin) splits the vertices into two disjoint sets $A$ and $... | davidlowryduda | 9,754 | <p>Roughly speaking, your result is that
$$ \sigma(n) \ll n \log n,$$
where I use $\ll$ as a sort of <a href="https://en.wikipedia.org/wiki/Big_O_notation" rel="nofollow">big Oh</a> notation. It's actually known that
$$ \sigma(n) \ll n \log \log n.$$
The difference between what you've shown and what is known is the dif... |
215,898 | <p>This is a quick follow up to my other <a href="https://mathematica.stackexchange.com/q/215884/44420">question</a> which I thought was different enough to warrant a separate post.</p>
<p><strong>My Question</strong></p>
<blockquote>
<p>How do you plot a region (such as <span class="math-container">$f(x,y)>z$</... | Rupesh | 63,381 | <p>You may also try Piecewise:
Something similar to this for defining Colorfunction of separate regions.</p>
<pre><code> colfn = Piecewise[{{Red, 3/2 > f[#, #2] > 1}, {White, True}}] &
Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2},
PlotPoints -> 200, PlotRange -> All, ColorFunction -> colfn,
... |
907,850 | <p>We have tiles of size <code>2 * 1</code>. We need to arrange the tiles to get the floor size of <code>m * n</code>.</p>
<p>For <code>m = 3</code>, we get arrangement like this:</p>
<p><img src="https://i.stack.imgur.com/SicS8.png" alt="enter image description here">
<br></p>
<p>My question is: In how many differe... | MHS | 158,061 | <p>There is a general formula for the number of valid domino tilings in an (m x n) rectangle, which was first found by Kastelyn (around 1960).</p>
<p>$
\prod_{j=1}^{\lceil\frac{m}{2}\rceil} \prod_{k=1}^{\lceil\frac{n}{2}\rceil} \left ( 4\cos^2 \frac{\pi j}{m + 1} + 4\cos^2 \frac{\pi k}{n + 1} \right )
$</p>
<p>This f... |
907,850 | <p>We have tiles of size <code>2 * 1</code>. We need to arrange the tiles to get the floor size of <code>m * n</code>.</p>
<p>For <code>m = 3</code>, we get arrangement like this:</p>
<p><img src="https://i.stack.imgur.com/SicS8.png" alt="enter image description here">
<br></p>
<p>My question is: In how many differe... | MHS | 158,061 | <p>Here is a link to a presentation (by some undergraduate student) about the computation of those numbers:</p>
<p><a href="http://math.cmu.edu/~bwsulliv/domino-tilings.pdf" rel="nofollow">http://math.cmu.edu/~bwsulliv/domino-tilings.pdf</a></p>
<p>It also contains references to Kastelyn's original paper and other in... |
1,833,854 | <p>We have $M$ Binomial random variables, where $X_0 \sim $ Bin$(n,p)$ and $X_i \sim $ Bin$(n,1/2)$. </p>
<p>Suppose $p > 1/2$. I'm interested in the probability that $\mathbb{P}(\max \{X_1,\dots,X_M\} \geq X_0)$. Is this tractable? </p>
<p>If not, is it tightly boundable/approximable? If this is a very difficult ... | Graham Kemp | 135,106 | <p>The maximum of $\{X_i\}_{i\in\{1;M\}}$ will be at least as great as $X_0$ when it is not that all values are less than $X_0$. (Employ the rule of complements.) Assuming that $X_0$ is <em>also</em> independent of the iid variables then:
$$\begin{align}
\mathsf P\left(\max_{i=1}^M\{X_i\} \geqslant X_0\r... |
2,748,495 | <p>I have a numeric table for artillery operations (Royal Italian Army, year 1940), in the instructions it refers to a measure of a planar angle as $32.00^{\circ\circ}$ and it seems to me that this angle is equivalent to $\pi$ radian. I had a look at <a href="https://en.wikipedia.org/wiki/Gradian" rel="nofollow norefer... | David K | 139,123 | <p>Working from context, in order for the angles to be measured as shown in the figures, the direction from $S$ to $D,$ measured by the goniometer at $S,$ must be $\pi$ radians.
Otherwise the angles $\alpha$ and $\beta$ would not be measured in the
indicated orientations from the same base line.</p>
<p>Therefore I wou... |
1,330,858 | <p>Suppose I'm at $(x=0,y=0)$ and I want to get to $(x=1,y=1)$. The shortest path is the diagonal and it has length $\sqrt{2}$. But what if I'm only allowed to make moves in coordinate directions---e.g., $1/2$ along $x$, $1/2$ along $y$, another $1/2$ along $x$, and a final $1/2$ along $y$. Then the length of my path i... | Zach L | 249,196 | <p>This is indeed because you cannot exchange the limit that you have described and the path length. No matter how large your n gets, if we look at one change in x and one change in y, the total distance from start to finish will always be √2 * the change in x (or y, because they are both the same). The error does not ... |
52,802 | <p>This is partly a programming and partly a combinatorics question.</p>
<p>I'm working in a language that unfortunately doesn't support array structures. I've run into a problem where I need to sort my variables in increasing order.</p>
<p>Since the language has functions for the minimum and maximum of two inputs (b... | Patrick Da Silva | 10,704 | <p>$e-A = A$ is clearly impossible... $e-A$ contains no element of the set $A$ by construction. The only thing that could define an identity is clearly the empty set, because $(e-A)$ doesn't contain any element of $A$, hence must not contain any element at all, for if $e$ contains an element, there exists a set $A$ whi... |
2,439,744 | <p>How to prove that $\lim_{x\to 0,y\to 0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$?</p>
<p>I already calculated that when a = 1 by multiplying $\sqrt {a+x^2y^2} + 1$ on both denominator and numerator and using the fact $x^2y^2<(x^2+y^2)^2/4$.</p>
<p>Any help will be appreciat... | Community | -1 | <p>If your function <em>can</em> have limit at $(0,0)$ for some $a_0$ then it can be defined at $a_0$ to be continuous at $a_0$. So suppose that we have some $a_0$ for which your function can be continuous at $a_0$.</p>
<p>If I recall corectly if $f$ is continuous then the limit is equal to iterated limits so we have ... |
2,662,605 | <p>Problem: If ${F_n}$ is a sequence of bounded functions from a set $D \subset \mathbb R^p$ into $ \mathbb R^q$ and if ${F_n}$ converges uniformly to $F$ on $D$, then $F$ is also bounded. </p>
<p>Proof(Attempt): Let $\epsilon >0$. Since ${F_n}$ converges uniformly to $F$ on $D$, then there is an $N \in\mathbb R$ s... | user284331 | 284,331 | <p>The assumption that each $F_{n}$ is bounded does not necessarily mean that there is an $M>0$ such that $|F_{n}(x)|\leq M$ for all $x\in D$ and $n=1,2,...$</p>
<p>Rather, we need to establish the uniform bound for all $F_{n}$. So the sequence is uniformly Cauchy, so $|F_{n}(x)-F_{m}(x)|<1$ for all $x\in D$ and... |
4,416,001 | <p>Given <span class="math-container">$f(xy) = f(x+y)$</span> and <span class="math-container">$f(11) = 11$</span>, what is <span class="math-container">$f(49)$</span>?</p>
| Maksim | 617,634 | <p><span class="math-container">$f(49)=f(0+49)=f(0\times49)=f(0)=f(11 \times 0)=f(11+0)=f(11)=11$</span></p>
|
2,709,273 | <blockquote>
<p>Determine the kernel of the following group homomorphism:
$$
\phi\colon\mathbb Z/270\mathbb Z\to\mathbb Z/270\mathbb Z\colon\overline x\mapsto\overline{6x}.
$$
Then find the solutions of the following system of equations in $\mathbb Z/270\mathbb Z$:
\begin{align}
6x=3\mod 27\\
6x=2\mod 10
\end{a... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">\begin{align}
6x &\equiv 3 \pmod {27}\\
2x &\equiv 1 \pmod{9} \\
\color{red}{x} &\color{red}{\equiv} \color{red}{5 \pmod{9}} \\
\hline
6x &\equiv 2 \pmod{10} \\
3x &\equiv 1 \pmod{5} \\
\color{red}x &\color{red}{\equiv} \color{red}{2 \pmod{5}} ... |
2,861,293 | <p>I found this statement with the proof:</p>
<p><a href="https://i.stack.imgur.com/bGRiZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bGRiZ.jpg" alt="enter image description here" /></a></p>
<p>But I don't understand the proof. Where is the contradiction? We have a nonempty interval <span class="... | Paul Frost | 349,785 | <p>In the above form the proof isn't done well. You know that $y_0^- < f(x_0)$ or $f(x_0) < y_0^+$. Hence at least one of the intervals $J^- = (y_0^-, f(x_0))$ and $J^+ = (f(x_0),y_0^+)$ is non-empty and does not contain any point of the form $f(x)$. This comes from the fact that $f(x_0)$ is contained in none of ... |
4,419,565 | <p>Consider the function <span class="math-container">$f(x)$</span> and let <span class="math-container">$g(x)=f(cx)$</span>.</p>
<p>By the definition of derivative</p>
<p><span class="math-container">$$f'(x)=\frac{df(x)}{dx}=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\tag{1}$$</span></p>
<p>and so the definition of <s... | epi163sqrt | 132,007 | <p>This is just a supplementary part to the nice answer of @KarlSchildkraut. I think it is helpful to consider the <em>complete</em> definition of functions and look at their relationship somewhat more detailed.</p>
<p><strong>The setting more detailed:</strong></p>
<p>We start with a differentiable function
<span clas... |
2,482,868 | <p>I am trying to find</p>
<p>$$ \lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}],$$</p>
<p>where $p>0$. I have tried to factor out as</p>
<p>$$(1+x^{p+1})^{\frac1{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} =x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... | user70925 | 479,902 | <p>Let try to use a general method to solve this kind of limits by
looking at the first order Taylor expansion of your expression:</p>
<p>\begin{aligned}\left( 1+x^{p+1}\right)^{\frac{1}{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} &=
x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... |
815,739 | <p>Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ?</p>
| fosco | 685 | <p>My two cents: to a certain extent your question is not silly, and can be partially answered.</p>
<p>Let's start unbiased: you have this big box where you put all vector spaces $\mathbb R^n$ and linear maps between them. If you think up to isomorphism it's a category, the category $\bf Vect$ of finite dimensional re... |
184,940 | <p>Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is embedded into space.</p>
<blockquote>
<p><strong>Question:</strong></p>
<p>is it also possible, to calculate the euclidean ... | Sebastian | 4,572 | <p>My answer is contained in the one of Thomas which was posted when I wrote my answer.</p>
<p>No, consider the plane $\{(x,y,0)\mid x,y\in\mathbb,0<y<2\pi\}$ and the subset
$\{(x,\cos y, \sin y)\mid x,y\in\mathbb,0<y<2\pi\}$ of the round cylinder in euclidean 3-space. Both are globally isometric, so all ... |
162,520 | <p>This is a cross-post from <a href="https://math.stackexchange.com/questions/738094/good-book-on-analytic-continuation">MSE</a>.</p>
<p>For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include ... | user74550 | 74,550 | <p>There is a book on Complex Variables with Physical Applications by Arthur A.Hauser, Jr.
is theory and step-by step solutions to 760 problems.Chapter 10 of this book deals at grant extend on Analytic Continuation.</p>
|
1,412,862 | <p>The author of my textbook has an unsatisfactory proof when it is describing the properties of the closure of a set.
I'm using <span class="math-container">$E^*$</span> for E closure. Also, <span class="math-container">$E'$</span> indicates the set of limit points of <span class="math-container">$E$</span>.</p>
<bloc... | Rupsa | 264,612 | <p>for your 2nd ques prove if F contains E then F' contains E' if E'=null set then there nothing to prove . Let E' is not a null set then x<em>e</em>E' implies x is a limit point of the set E in X implies E meet [B(x,r){x}] is not equal to null set for all +ve real r implies ... |
1,992,789 | <blockquote>
<p>Let $\{x_n\}$ be a sequence that does not converge and let L be a real
number. Prove that there exist $\epsilon >0$ and a sub-sequence
$\{x_{p_n}\}$ of $\{x_n\}$ such that $|x_{p_n}-L|>\epsilon$ for all n.</p>
</blockquote>
<p>I don't have any idea on how to prove this. Any advice and sugge... | barak manos | 131,263 | <p>Use <em>inclusion/exclusion</em> principle:</p>
<ul>
<li>Include the total number of sequences, which is $\frac{8!}{2!2!2!2!}$</li>
<li>Exclude the number of sequences containing $11$, which is $\frac{7!}{1!2!2!2!}$</li>
<li>Exclude the number of sequences containing $22$, which is $\frac{7!}{1!2!2!2!}$</li>
<li>Ex... |
1,027,235 | <p>Let N = 12345678910111213141516171819. How can I use modular arithmetic to show that N is (or isn't) divisible by 11? In general, how can I apply modular arithmetic to determine the divisibility of an integer by a smaller integer? I am finding modular arithmetic very confusing and unintuitive. I can understand "simp... | Silvio Moura Velho | 74,553 | <p>$N = a,bcd$; $a' \equiv (− cd \mod 11 + a) \mod 11 \rightarrow a'b$</p>
<p>$N = bcd;$ $a' \equiv (−cd \mod 11 + 0) \mod 11 \rightarrow a'b$</p>
<p>If $11 |a 'b$ then $11|N$</p>
<p>Apply the algorithm repetitively from right to left, always eliminating the last two digits.If the result is a multiple of $11$ then t... |
454,426 | <blockquote>
<p>In set theory and combinatorics, the cardinal number $n^m$ is the size of the set of functions from a set of size m into a set of size $n$.</p>
</blockquote>
<p>I read this from this <a href="http://en.wikipedia.org/wiki/Empty_product#0_raised_to_the_0th_power" rel="nofollow noreferrer">Wikipedia pag... | Stefan Hamcke | 41,672 | <p>An $m$-digit number $x_1 x_2 ... x_m$ where each $x_i$ can be a digit from the set $\{1,...,n\}$ is basically the same as a function from $M:=\{1,2,...,m\}$ to $N:=\{1,2,...,n\}$. Let $x:M\to N$ be such a map. Then you can construct the number $x(1)x(2)...x(m)$. If on the other hand you have a number $x_1 x_2 ... x_... |
501,250 | <p>I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ and $\textbf{y}=(y_1,y_2) \in \text R^2$. If true, I'd also like to extend this to $\textbf{x} \in \text R^{n_1+n_2}$... | Arash | 92,185 | <p>The set $C =\{ \frac{1}{n}|n \in \mathbb{N} \}$ is not open in $\mathbb R$. The reason is that for each $x\in C$ you cannot find a neighborhood of $x$ such that $N(x,r)\subset C$. For each neighborhood of $x=\frac{1}{n}$, it is possible to find $y$ such that $\frac{1}{n+1}<y<\frac{1}{n}$ which means that $y\n... |
1,105,126 | <p><img src="https://i.stack.imgur.com/XtrB7.png" alt="enter image description here"></p>
<p>My attempt at the solution is to let P(n) be $10^{3n} + 13^{n+1}$</p>
<p>P(1)= $10^3 + 13^2 = 1169$</p>
<p>Thus P(1) is true.</p>
<p>Suppose P(k) is true for all $k \in N$
$\Rightarrow P(k) = 10^{3k} + 13^{k+1} = 10^{3k} +... | orangeskid | 168,051 | <p>Consider $P$ the amount of papers. In one minute the first person delivers $\frac{P}{40}$, the second one $\frac{P}{50}$. Together in one minute they deliver $\frac{P}{40}+\frac{P}{50}$. So they need </p>
<p>$$\frac{P}{ \frac{P}{40}+\frac{P}{50}}$$ minutes to deliver the whole thing. </p>
<p>The quantity $P$ magic... |
1,521,649 | <p>I need to show that at most finitely many terms of this sequence are greater than or equal to $c$. </p>
<p>I don't know if it is the wording of the problem but I don't know what this is asking me to do. Help on this would be amazing! And thank you in advance.</p>
| egreg | 62,967 | <p>I guess you have already proved the main property
$$
\ln(xy)=\ln x+\ln y
$$
(for positive reals $x$ and $y$). By an easy induction you get that
$$
\ln(a^m)=m\ln a
$$
for a positive integer $m$. If $m<0$, you have
$$
\ln(a^m)=\ln\frac{1}{a^{-m}}=-\ln(a^{-m})=-(-m\ln a)=m\ln a
$$</p>
<p>Now suppose $b=m/n$, where,... |
4,292,815 | <p>Compute line integral <span class="math-container">$\int_a^b (y^2z^3dx + 2xyz^3dy + 3xy^2z^2dz)$</span> where <span class="math-container">$a = (1,1,1)$</span> and <span class="math-container">$b = (2,2,2)$</span></p>
<p>What I have done:</p>
<p>To find <span class="math-container">$t$</span> I used the calculation ... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$(x,y,z)=(t+1,t+1,t+1)$</span> and you want to go from <span class="math-container">$(1,1,1)$</span> to <span class="math-container">$(2,2,2)$</span>, you should compute that integral with <span class="math-container">$a=0$</span> and <span class="math-container">$b=1$</span>.</p>
... |
858,353 | <p>$$x^2(y')^2+3xyy'+2y^2=0$$
I have no idea how to start, I probably need to do some tricky substitution but as of now I cant see any options.</p>
| johannesvalks | 155,865 | <p>Hint:</p>
<p>$$
\Big( x y' + \frac{3}{2} y \Big)^2 = x^2 (y')^2 + 3xy y' + \frac{9}{4} y^2
$$</p>
|
3,489,280 | <p>If <span class="math-container">$f_n → f$</span> and <span class="math-container">$g_n → g$</span>, does <span class="math-container">$f_n g_n → fg$</span> in the space <span class="math-container">$C[0, 1]$</span> for the norms <span class="math-container">$||.||_1$</span> and <span class="math-container">$||.||_∞$... | zhw. | 228,045 | <p>Unfortunately, both of your guesses are incorrect.</p>
<p>To see the <span class="math-container">$\|\,\|_1$</span> result fails, let</p>
<p><span class="math-container">$$f_n(x)= \frac{1}{[\ln (n+1)(x+1/n)]^{1/2}}.$$</span></p>
<p>Verify that <span class="math-container">$f_n\to 0$</span> in the <span class="mat... |
3,489,280 | <p>If <span class="math-container">$f_n → f$</span> and <span class="math-container">$g_n → g$</span>, does <span class="math-container">$f_n g_n → fg$</span> in the space <span class="math-container">$C[0, 1]$</span> for the norms <span class="math-container">$||.||_1$</span> and <span class="math-container">$||.||_∞$... | mechanodroid | 144,766 | <p>For a counterexample to the <span class="math-container">$\|\cdot\|_1$</span> norm, consider
<span class="math-container">$$f_n(t) = \begin{cases} n-n^3t, &\text{ if } t \in \left[0,\frac1{n^2}\right]\\
0, &\text{ if } t \in \left[\frac1{n^2},1\right]\\
\end{cases}$$</span></p>
<p>Then <span class="math-con... |
1,914,752 | <p>dividing by a whole number i can describe by simply saying split this "cookie" into two pieces, then you now have half a cookie. </p>
<p>does anyone have an easy way to describe dividing by a fraction? 1/2 divided by 1/2 is 1</p>
| John Joy | 140,156 | <p>Dividing a whole number (lets say one) into two pieces can be restated as "If one (lets say gallon) fills two containers, then how much does one container hold?". Of course the answer is $\frac{1}{2}$ gal.</p>
<p>Using that same language consider "If $\frac{5}{16}$ Gal. fills $\frac{3}{7}$ of a container, then how ... |
1,479,822 | <p>$f\cdot g$ is Lebesgue integrable, g is Lebesgue integrable, can we deduce that f is Lebesgue integrable? </p>
<p>$f\cdot g$ is integrable, g is integrable, can we deduce that f is finite a.e.? </p>
| Chappers | 221,811 | <p>If the support of $g$ and the support of $f$ have intersection with measure zero, you can't conclude anything.</p>
|
203,827 | <p>Suppose I have the following lists: </p>
<pre><code>prod = {{"x1", {"a", "b", "c", "d"}}, {"x2", {"e", "f",
"g"}}, {"x3", {"h", "i", "j", "k", "l"}}, {"x4", {"m",
"n"}}, {"x5", {"o", "p", "q", "r"}}}
</code></pre>
<p>and </p>
<pre><code>sub = {{"m", "n"}, {"o", "p", "r", "q"}, {"g", "f", "e"}};
</code><... | Chris Degnen | 363 | <pre><code>getMatches[prod_, sub_] := Module[{test},
Scan[(test[Sort[#]] = True) &, sub];
Cases[prod, {_, y_?test}]]
getMatches[prod, sub]
</code></pre>
<blockquote>
<p>{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}</p>
</blockquote>
<p>Also</p>
<pre><code>getMatches[prod_, su... |
1,997,513 | <p>I spend so much time for proving this triangle and i still don't know. </p>
<p>Question :</p>
<p>Given Triangle ABC, AD and BE are altitudes of the triangle. Prove that Triangle DEC similarity with triangle ABC</p>
| Vidyanshu Mishra | 363,566 | <p>Construct a figure from the data you have given, now follow these steps(you just have to prove equality of any two angles of respective triangles):</p>
<p>Since BE and AD are perpendiculars so you get ∠BEA = ∠ADB =90 Degrees.Now you can see that ABDE is a cyclic quadrilateral {as ∠BEA = ∠ADB(you can call them angl... |
4,263,631 | <p>so i have question about existence of function <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> such <span class="math-container">$f$</span> is not the pointwise limit of a sequence of continuous functions <span class="math-container">$\mathbb{R} \to \mathbb{R}$</span>.</p>
<p>i'm created a family o... | Mark | 470,733 | <p>You are close. Measure of a set is equal to the Lebesgue integral of the constant function <span class="math-container">$1$</span> on that set. So the measure of your set is equal to the double integral <span class="math-container">$\iint\limits_A 1 dxdy$</span>. Since the function has an absolutely convergent impro... |
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | Andrea Mori | 3,602 | <p>As for your first question: if you think of your elliptic curves as plane cubics (Weierstrass'model) the isomorphism between them is a polynomial function. Polynomials include only finitely many coefficients and the isomorphism is defined over the field generated by them, which is finite over $\Bbb Q$.</p>
|
497,546 | <p>Let $A$ be an infinite set.</p>
<p>Then, we can construct an injective function $f:\omega \rightarrow A$. </p>
<p>But how do i construct this via orginal statement of $AC_\omega$? (i.e. $\forall countable X, [\emptyset \notin X \Rightarrow \exists f:X\rightarrow \bigcup X \forall A\in X, f(A)\in A$)</p>
<p>So my ... | Herman Jaramillo | 62,024 | <p>In general, a generating function for a sequence of functions $P_n(x)$, is a function
$G(x,t)$, such that</p>
<p>\begin{eqnarray*}
G(x,t)= \sum_{n=0}^{\infty} P_n(x) t^n,
\end{eqnarray*}
where, by matching equal powers of $t$, the Taylor series expansion of $G(x,t)$ provides
the functions $P_n(x)$. In particular ... |
4,258,226 | <p>Suppose a group <span class="math-container">$G$</span> splits as a semidirect product <span class="math-container">$N\rtimes\mathbb{Z}_2$</span>, and let <span class="math-container">$\phi:G\to\mathbb{Z}_2$</span> the the associated quotient map. If I have a subset of elements <span class="math-container">$\{g_1,\d... | Adayah | 149,178 | <p>No, you can't.</p>
<p>Consider <span class="math-container">$G = \mathbb{Z}^2 \rtimes \mathbb{Z}_2$</span> where <span class="math-container">$1 \in \mathbb{Z}_2$</span> corresponds to the automorphism <span class="math-container">$\varphi \in \operatorname{Aut}(\mathbb{Z}^2, +)$</span> such that <span class="math-c... |
893,822 | <p>If $p(x)$ has integer coefficients and $p(100)$ equals $100$ what is the maximum number of integer solutions $k$ to the equation $p(k)=k^3$.</p>
<p>I have tried hard to solve this problem but I could not figure it out. I tried some particular cases but got nowhere, could someone please show me how to get the answer... | ShakesBeer | 168,631 | <p>The easy way:
<br>
Let $f(x)=RHS-LHS$.
<br>
$f'(x)=2n^2(1+n(x-1))^{n-1}-n(1+\frac{x}{\sqrt{x^2-1}})(x+\sqrt{x^2-1})^{n-1}-n(1-\frac{x}{\sqrt{x^2-1}})(x-\sqrt{x^2-1})^{n-1}$
<br><br>
$f'(x)=2n^2(1+n(x-1))^{n-1}-\frac{n}{\sqrt{x^2-1}}(x+\sqrt{x^2-1})^{n}+\frac{n}{\sqrt{x^2-1}}(x-\sqrt{x^2-1})^{n}$
<br>
$f'(x) > 2n... |
3,878,380 | <p>Do the columns of a matrix always represent different vectors? If so, I don't understand how if I have a <span class="math-container">$3\times3$</span> matrix where the rows represent the dimensions and I multiply it by a <span class="math-container">$3\times1$</span> column vector with the same dimensions, it will ... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">$$\left(\begin{array}{c}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right)
\cdot
\left( \begin{array}{c}
x \\ y \\ z
\end{array}
\right)
=
x\left( \begin{array}{c}
a \\ d \\ g
\end{array}
\right) +
y\left( \begin{array}{c}
b \\ e \\ h
\end{array}
\ri... |
175,971 | <p>Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.</p>
<p>Thanks a lot!</p>
| Axel Boldt | 133,539 | <p>When people in modern algebraic geometry talk about <span class="math-container">$\operatorname{Spec}(k)$</span>, the prime spectrum of a field <span class="math-container">$k$</span>, they mean it in the sense of schemes.</p>
<p>Every scheme has an underlying topological space; in the case of <span class="math-cont... |
1,132,599 | <p>For which prime numbers p does the congruence $x^2+x+1\equiv0$ mod p have solutions? </p>
<p>I am new to the topic of quadratic reciprocity and I know how to answer this question had it been for which prime numbers p does the congruence $x^2\equiv-6$ mod p have solutions? </p>
<p>Can I perhaps split the congruence... | David | 119,775 | <p><strong>Hint</strong>. For $p\ne2$ we have
$$x^2+x+1\equiv0\pmod p\quad\Leftrightarrow\quad
(2x+1)^2\equiv-3\pmod p\ .$$</p>
|
2,041,484 | <p>Solve the system of equations for all real values of $x$ and $y$
$$5x(1 + {\frac {1}{x^2 +y^2}})=12$$
$$5y(1 - {\frac {1}{x^2 +y^2}})=4$$</p>
<p>I know that $0<x<{\frac {12}{5}}$ which is quite obvious from the first equation.<br>
I also know that $y \in \mathbb R$ $\sim${$y:{\frac {-4}{5}}\le y \le {\frac 4... | Stefan4024 | 67,746 | <p>Let $z=x + iy \in \mathbb{C}$, now for $|z|^2 = x^2+y^2 \not = 0$ have that:</p>
<p>$$\left(5x + \frac{5x}{x^2+y^2}\right) + i\left(5y - \frac{5y}{x^2+y^2}\right) = 12 + 4i$$</p>
<p>$$5(x+iy) + \frac{5(x-iy)}{x^2+y^2} = 12 + 4i$$</p>
<p>$$5(x+iy) + \frac{5}{x+iy} = 12 + 4i$$</p>
<p>$$5z + \frac{5}{z} = 12 + 4i$$... |
739,516 | <p>Is there a closed form for $$\int_{0}^{\infty}e^{ax^3+bx^2}\,\mathrm{d}x $$?</p>
| Sasha | 11,069 | <p>Let $a<0$, and changing variable $y = (-a x^3)$:
$$
\int_0^\infty \exp(a x^3 + b x^2) \mathrm{d}x = \alpha\int_0^\infty \exp\left(-y + \beta y^{2/3} \right) y^{-2/3} \mathrm{d}y
$$
where $\alpha = 1/\left(3 (-a)^{1/3}\right)$ and $\beta = b (-a)^{-2/3}$.</p>
<p>Now, using $y^{-2/3} \exp\left(\beta y^{2/3}\rig... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Ayman Hourieh | 4,583 | <p>$f(x)$ is separable since its derivative is $f'(x) = -1 \ne 0$.</p>
<p>Suppose $\theta$ is a root of $f(x) = x^p - x + a$. Using the Frobenius automorphism, we have:
\begin{align}
f(\theta + 1) &= (\theta + 1)^p - (\theta + 1) + a\\
&= \theta^p + 1^p - \theta - 1 + a\\
&= \theta^p - \theta + a\\
&= ... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Or Kedar | 460,215 | <p>A little bit late, but I have no doubt people will still come back to this question.
To the proof:</p>
<p>If <span class="math-container">$r$</span> is a root of <span class="math-container">$f$</span> then <span class="math-container">$r+j$</span> is a root of <span class="math-container">$f$</span>, for <span clas... |
1,056,041 | <p>Look at problem 8 :</p>
<blockquote>
<p>Let $n\geq 1$ be a fixed integer. Calculate the distance:
$$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over
polynomials with degree less than $n$ with real coefficients and $f$
runs over functions $$ f(x)=\sum_{k=n}^{+\infty}c_k\, x^k$$ defined on
the c... | Jimmy R. | 128,037 | <p>Your approach (although nice) has a flaw in the second bullet. The problem is that there you count two different things: on the one hand ways to choose a box and on the other hand ways to choose a ball and this results to a confusion. In detail</p>
<ol>
<li>Your denominator is correct,</li>
<li>Your numerator is mi... |
2,663,130 | <p>Let $f:\mathbb{R}^2\to \mathbb{R}^2$ be function $f(x,y)=(\frac{1}{2}x+y,x-2y)$. Find a image of set $A\subset\mathbb{R}^2$ bounded with lines $x-2y=0, x-2y+2=0, x+2y-2=0, x+2y-3=0.$</p>
<p>Set $A$ is parallelogram with vertices $(1,\frac{1}{2}), (\frac{3}{2},\frac{3}{4}), (\frac{1}{2},\frac{5}{4}), (0,1)$.</p>
<p... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Note that</p>
<p>$$\frac{a_{n+1}}{a_n}=\frac{(n+1)^2}{n(n+2)}=\frac{n^2+2n+1}{n^2+2n}=1+\frac{1}{n^2+2n}>1$$</p>
<p>thus $a_n$ is <strong>strictly increasing</strong>.</p>
|
2,953,757 | <p>My try to solve this question goes as follows:</p>
<p><span class="math-container">$g=gcd(n^2+1, (n+1^2)+1) = gcd(n^2+1, 2n+1) = gcd(n^2-2n, 2n+1)$</span>.</p>
<p>By long division: </p>
<p><span class="math-container">$$n^2-2n = -2n(2n+1) + 5n^2$$</span></p>
<p>Since <span class="math-container">$g$</span> divid... | Bill Dubuque | 242 | <p>mod <span class="math-container">$\, (f(n),\!\!\overbrace{n^2\!+\!1}^{\color{#c00}{\Large n^2\ \equiv\ -1}}\!\!\!)\!:\ \ 0\equiv \overbrace{\color{#c00}{n^2}\!+\!2n\!+\!2}^{\large f(n)}\equiv 2n\!+\!1\ \Rightarrow\ 0\equiv \overbrace{(1\!+\!2n}^{\Large \color{#0a0}w})(\overbrace{1\!-\!2n}^{\Large\color{#0a0}{ \ba... |
3,066,913 | <p>Given a linear operator <span class="math-container">$T$</span> on a finite-dimensional vector space <span class="math-container">$V$</span>, satisfying <span class="math-container">$T^2 = T$</span> answer the following:</p>
<p>(a) Using the dimension theorem, show that <span class="math-container">$N(T) \bigoplus ... | Community | -1 | <p>Let <span class="math-container">$ST(X)=X$</span> for any <span class="math-container">$X \in \mathbb{R^n}$</span>. Since <span class="math-container">$S, T$</span> are linear and these have inverses, then we have <span class="math-container">$T^{-1}S^{-1}$</span> is linear an exist so</p>
<p><span class="math-cont... |
3,596,514 | <p>Let <span class="math-container">$0\le x \le 1$</span>
find the maximum value of <span class="math-container">$x(9\sqrt{1+x^2}+13\sqrt{1-x^2})$</span></p>
<p>I try to use am-gm inequality to solve this because it's similar to CMIMC2020 team prob.12 but i don't know how to do next. </p>
| Michael Rozenberg | 190,319 | <p>By C-S and AM-GM we obtain:
<span class="math-container">$$x\left(9\sqrt{1+x^2}+13\sqrt{1-x^2}\right)\leq x\sqrt{(27+13)(3(1+x^2)+13(1-x^2))}=$$</span>
<span class="math-container">$$=\sqrt{16\cdot5x^2(8-5x^2)}\leq\sqrt{16\left(\frac{5x^2+8-5x^2}{2}\right)^2}=16.$$</span>
The equality occurs for <span class="math... |
352,305 | <p>For <span class="math-container">$\mathcal{S}$</span> the <span class="math-container">$(\infty,1)$</span>-category of spaces <a href="https://mathoverflow.net/questions/239383/the-homotopy-category-is-not-complete-nor-cocomplete">its homotopy category <span class="math-container">$h\mathcal{S}$</span> does not have... | Kevin Arlin | 43,000 | <p>As has already been said, the homotopy category does not admit filtered colimits in general, but it’s much worse than that. Even colimits in an <span class="math-container">$\infty$</span>-category which don’t give rise to colimits in the homotopy category sometimes do give rise to <em>weak</em> colimits. (A weak co... |
744,787 | <p>I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line
$$
\frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}.
$$</p>
<p>I think there must bee some kind of rule to do this, but I can't find it anyway.</p>
| MarkisaB | 139,345 | <p>Imagine you put point A in a plane that is perpendicular to given line, then the line you're looking for be in this plane, and could be found using the plane line intersection point and point A.
<img src="https://i.stack.imgur.com/S1uzQ.png" alt="enter image description here"></p>
<p>something like this. </p>
<ol>... |
285,719 | <p>Why is $(-3)^4 =81$ and $-3^4 =-81 $?This might be the most stupidest question that you might have encountered,but unfortunately i'am unable to understand this.</p>
| P.K. | 34,397 | <p>$$(-3)^4 = -3 \cdot -3\cdot -3\cdot -3 = 9\cdot9=81 $$Now, when we have to "simplify" the expression $-3^4$, first we do the exponent and then the rest. So this simplifies to $-(81) = -81$</p>
<hr>
<p>This question had confused me a lot too, and then I got my enlightenment (now waiting for the badge)...</p>
|
3,456,351 | <pre><code>Apples: 1
Apple Value: 2500
Pears: lowest = 1, highest = 10
</code></pre>
<p>If I have 1 apple and my apple is worth <strong>2500</strong> if I have 1 pear, how can I calculate the value of my apple if I have X pears, at a maximum of 10 pears and a minimum of 1 pear, where 1 pear represents 100% value and 1... | John Omielan | 602,049 | <p>The identity is true for <span class="math-container">$n = 0$</span>, so consider <span class="math-container">$n \ge 1$</span>. Also, let</p>
<p><span class="math-container">$$m = \sqrt{n} + \sqrt{n + 1} \tag{1}\label{eq1A}$$</span></p>
<p>With <span class="math-container">$m \gt 0 \; \to \; m = \sqrt{m^2}$</span>,... |
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | Christian Blatter | 1,303 | <p>An "equality by definition" is a directed mental operation, so it is <em>nonsymmetric</em> to begin with. It's only natural to express such an equality by a nonsymmetric symbol such as $:=\, .\ $ Seeing a formula like $e=\lim_{n\to\infty}\left(1+{1\over n}\right)^n$ for the first time a naive reader would look for a... |
2,696,579 | <p>So I understand that in order to have a particular solution you have to have a non homogenous second order differential equation. However I have a slightly difficult time comprehending how to pick the particular solution given $g(t)$. </p>
<p>The reason I ask is our teacher skimmed over it and hardly covered it in ... | Martin Argerami | 22,857 | <p>Since $f$ is monotonic, it is continuous almost everywhere. Let $x_0$ be a point such that $f$ is continuous at $x_0$. Then, noting that $f(0)=0$ and $-f(x_0)=f(-x_0)$,
$$
\lim_{h\to0}f(h)=\lim_{h\to0}f(x_0+h-x_0)=\lim_{h\to0} f(x_0+h)-f(x_0)=f(x_0)-f(x_0)=0.
$$
So $f$ is continuous at $0$. But then $f$ is continu... |
214,475 | <p>Function:
<a href="https://i.stack.imgur.com/sH7mh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sH7mh.png" alt="enter image description here"></a></p>
<p>I am to solve for <span class="math-container">$T_{12}(4.8), T_{24}(1.2)$</span>, using <strong>If</strong> and <strong>Which</strong> funct... | Vasily Mitch | 53,855 | <p>Try</p>
<pre><code>T[0] = 1;
T[1] = x;
T[n_] := (1/x) T[n - 2] - (2/7) T[n - 1];
</code></pre>
|
2,909,626 | <p>I'm having trouble remembering how to solve when you have an equation such as </p>
<p>$$0.3=(1-0.63x)^{5.26}$$</p>
<p>any help would be appreciated.</p>
| Joseph Adams | 583,129 | <p>Try thinking about it this way: your functional is a linear continuous function from the step functions to $\mathbb{R}$. Because of the linearity the continuity of the functional is equivalent to its uniform continuity. Now it is well known that a uniformly continuous function can be uniquely extended (as a uniforml... |
413,882 | <p>Let $\mathbb F$ be a field and $\mathbb F[x]$ the ring of polynomials with coefficients in $\mathbb F$. Let $p(x)$ be an irreducible polynomial in $\mathbb F[x]$. Let $k$ be a positive integer and consider the vector space $V$, over the field $\frac{\mathbb F[x]}{(p(x))}$with basis </p>
<p>$$1, p(x), p(x)^2, \ldot... | Konstantin Ardakov | 119,170 | <p>See <a href="https://math.stackexchange.com/questions/416688/isomorphic-rings-or-not/628040#628040">this follow-up question</a> for a discussion of <em>coefficient fields</em>.</p>
<p>By the proof of Cohen's Structure Theorem, there is a coefficient field inside $A = \mathbb{F}[X]/(p^k)$, hence an injective map $L ... |
10,622 | <p>Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t.</p>
<p>$\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i \subset K_{i+1} ^{\circ}$
</p>
<p>We can imagine that different exhaustions by compact sets 'propagate' at di... | Nate Eldredge | 822 | <p>Yes. Let $\{L_i\}_{i=1}^\infty$, $\{K_i\}_{i=1}^\infty$ be two exhaustions. It follows that $\bigcup_{i=1}^\infty K_i^\circ = U$, so for each $n$, $\{K_i^\circ\}_{i=1}^\infty$ is an open cover of $L_n$. By compactness, it has a finite subcover $\{K_{i_1}^\circ, \dots, K_{i_l}^\circ\}$. But if $m = \operatorname{... |
4,058,319 | <p>My background in mathematical logic, model theory, etc. is patchy, so I'm looking for a clearer way to think about this. (Edit: I'm not asking what an isomorphism is, I'm asking how to formalize the idea of "preserving all logical properties" in order to state the principle described below in its full powe... | Rob Arthan | 23,171 | <p>If <span class="math-container">$\mathbf{M_1}$</span> and <span class="math-container">$\mathbf{M_2}$</span> are structures for some signature <span class="math-container">$\Sigma$</span> (so <span class="math-container">$\Sigma = (<)$</span>) in your example. Then, by definition, <span class="math-container">$\m... |
3,166,419 | <blockquote>
<p>A fair coin is tossed three times in succession. If at least one of
the tosses has resulted in Heads, what is the probability that at
least one of the tosses resulted in Tails?</p>
</blockquote>
<p>My argument and answer: The coin was flipped thrice, and one of them was heads. So we have two unkn... | Graham Kemp | 135,106 | <blockquote>
<p>The coin flips are all independent of each other, </p>
</blockquote>
<p>Here's the error in your reasoning.</p>
<p>The coin flips are not <em>conditionally</em> independent under constraint that <em>at least one</em> is a head. </p>
<hr>
<p>Let <span class="math-container">$H_k$</span> be the eve... |
2,650,634 | <p>EDIT: I know how to integrate the last part. I'm just try to find mistake in converting Sum to integral</p>
<p>Question: </p>
<blockquote>
<p>$$a_n=\left(\left(1+\left(\frac1n\right)^2\right)\left(1+\left(\frac2n\right)^2\right)\cdots\left(1+\left(\frac{n}n\right)^2\right)\right)^n$$ find<br>
$$\lim_{n\to\inf... | hamam_Abdallah | 369,188 | <p>You are correct but..</p>
<p>By parts,</p>
<p>$$\int_0^1\ln (1+x^2)dx=$$
$$[x\ln (1+x^2)]_0^1-2\int_0^1\frac {x^2+1-1}{1+x^2}dx =$$</p>
<p>$$\ln (2)-2+2\arctan (1) =$$
$$\ln (2)-2+\frac {\pi}{2} =-\ln y$$</p>
|
3,148,094 | <p>In class, my professor computed the density of a gamma random variable by taking the derivative of its CDF, but he skipped many steps. I am trying to go through the derivation carefully but cannot reproduce his final result.</p>
<p>Let <span class="math-container">$k$</span> be the shape and <span class="math-conta... | Parcly Taxel | 357,390 | <p>Simple manipulations will give us inequalities like <span class="math-container">$A$</span>:</p>
<ul>
<li>Since <span class="math-container">$2<x$</span>, <span class="math-container">$x^2>4$</span> and <span class="math-container">$x^2-4>0$</span>.</li>
<li>Since <span class="math-container">$x\le 3$</spa... |
241,210 | <p>I am confused with the concept of topology base. Which are the properties a base has to have?</p>
<p>Having the next two examples for $X=\{a,b,c\}$:</p>
<p>1) $(X,\mathcal{T})$ is a topological space where $\mathcal{T}=\{\emptyset,X,\{a\},\{b\},\{a,b\}\}$. Which is the general procedure to follow in order to get a... | Brian M. Scott | 12,042 | <p>Let $X$ be a non-empty set. A collection $\mathscr{B}$ of subsets of $X$ is a <em>base</em> for some topology on $X$ if it satisfies two conditions:</p>
<ol>
<li>$\mathscr{B}$ covers $X$. That is, every point of $X$ belongs to at least one member of $\mathscr{B}$. </li>
<li>If $B_1,B_2\in\mathscr{B}$ and $x\in B_1... |
141,522 | <p>First a summary of the general problem I'm trying to solve:
I want to get <strong>a</strong> set of inequalities for a very complex function (If you are interested is the no-arbitrage conditions Black-Scholes equation with a volatility given by an SVI function)</p>
<p>So basically I'm trying to find the parameters ... | mikado | 36,788 | <p>One approach is to eliminate x using <code>Minimize</code>. Additional constraints can be applied </p>
<pre><code>f = a*b*(1 + x^2);
min = Minimize[f, x];
</code></pre>
<p>The minimum can be compared with the threshold and solved</p>
<pre><code>Reduce[First[min] > 2, {a, b}]
(* (a < 0 && b < 2/a... |
141,522 | <p>First a summary of the general problem I'm trying to solve:
I want to get <strong>a</strong> set of inequalities for a very complex function (If you are interested is the no-arbitrage conditions Black-Scholes equation with a volatility given by an SVI function)</p>
<p>So basically I'm trying to find the parameters ... | user64494 | 7,152 | <p>You wrote "I found of the function ForAll which solves my simple example, but doesn't work for me on the actual problem as I have also conditions on x (Is there any similar command with no such conditions?) ". </p>
<p>The <code>ForAll</code> function works with restricted variables too. How about the following?</p>... |
3,424,720 | <p>I want to calculate the above limit. Using sage math, I already know that the solution is going to be <span class="math-container">$-\sin(\alpha)$</span>, however, I fail to see how to get to this conclusion.</p>
<h2>My ideas</h2>
<p>I've tried transforming the term in such a way that the limit is easier to find:
... | Bernard | 202,857 | <p>Don't make things more complicated than they are. This quotient is the rate of variation of the function of <span class="math-container">$x$</span>: <span class="math-container">$\cos(\alpha +x)$</span>, hence its limit is the derivative of the function at <span class="math-container">$x=0$</span>.</p>
|
3,999,325 | <p>Let me start with some objects. Consider the <span class="math-container">$\mathrm{C}^*$</span>-algebra <span class="math-container">$A$</span> defined by:
<span class="math-container">$$A=M_1(\mathbb{C})\oplus M_2(\mathbb{C})\subset B(\mathbb{C}^3).$$</span>
Let <span class="math-container">$x=\mathbb{C}^3$</span> ... | Ruy | 728,080 | <p>If <span class="math-container">$\pi :A\to B(H)$</span> is any non-degenerate representation of the C*-algebra <span class="math-container">$A$</span>, and if <span class="math-container">$x$</span> is a unit vector in <span class="math-container">$H$</span>, then the state
<span class="math-container">$$
\varphi... |
109,961 | <p>Suppose $x^2\equiv x\pmod p$ where $p$ is a prime, then is it generally true that $x^2\equiv x\pmod {p^n}$ for any natural number $n$? And are they the only solutions?</p>
| Lubin | 17,760 | <p>The English word “any” is slippery, but I think that the question asks whether if $x\equiv x^2 \pmod{p}$, then $x\equiv x^2 \pmod{p^n}$ is true for all $n$. One possible way to look at a congruence that’s true modulo all powers of a prime $p$ is that it’s really a statement about equality in the field ${\mathbb Q}_p... |
4,011,864 | <p><span class="math-container">$$\lim_{n \to \infty}(3^n+1)^{\frac{1}{n}}$$</span></p>
<p>I'm fairly sure I can't bring the limit inside the 1/n and I don't think I can use l'Hôpital's rule. I'm pretty sure I'm meant to use the sandwich theorem but I'm not quite sure how to do that in this circumstance.</p>
| DatBoi | 734,160 | <p><span class="math-container">$$\lim_{n \to \infty}(3^n+1)^{\frac{1}{n}}=\lim_{n \to \infty}3(1+\frac{1}{3^n})^{\frac{1}{n}}=\lim_{n \to \infty}3e^{\frac{1}{3^nn}}=3$$</span></p>
|
4,011,864 | <p><span class="math-container">$$\lim_{n \to \infty}(3^n+1)^{\frac{1}{n}}$$</span></p>
<p>I'm fairly sure I can't bring the limit inside the 1/n and I don't think I can use l'Hôpital's rule. I'm pretty sure I'm meant to use the sandwich theorem but I'm not quite sure how to do that in this circumstance.</p>
| QC_QAOA | 364,346 | <p>We have</p>
<p><span class="math-container">$$3= (3^n)^{1/n}\leq (3^n+1)^{1/n}\leq (3^n+3^n)^{1/n}=(2\cdot 3^n)^{1/n}=3\sqrt[n]{2}$$</span></p>
<p>Thus</p>
<p><span class="math-container">$$3\leq \lim_{n\to\infty}(3^n+1)^{1/n}\leq \lim_{n\to\infty}3\sqrt[n]{2}=3$$</span></p>
<p>We conclude the limit is <span class="... |
621,109 | <p>I need to find all the numbers that are coprime to a given $N$ and less than $N$.
Note that $N$ can be as large as $10^9.$ For example, numbers coprime to $5$ are $1,2,3,4$.</p>
<p>I want an efficient algorithm to do it. Can anyone help? </p>
| mathlove | 78,967 | <p>What you want is <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function" rel="noreferrer">Euler's totient function</a>.
You'll find a formula there.</p>
|
621,109 | <p>I need to find all the numbers that are coprime to a given $N$ and less than $N$.
Note that $N$ can be as large as $10^9.$ For example, numbers coprime to $5$ are $1,2,3,4$.</p>
<p>I want an efficient algorithm to do it. Can anyone help? </p>
| Sabareesh Muralidharan | 241,483 | <p>First find the factors of it by factorisation
second get the prime numbers involved in that say $(a,b,\ldots)$.
E.g., $18=(2^1)(3^2)$. The prime factors are $2$ and $3$.
Third formula:
$$\text{No. of coprimes to $N$}= N(1-1/a)(1-1/b)\cdots$$</p>
|
102,624 | <p>I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one. </p>
<p>Since Maass cusp forms always are defined for a Fuchsian lattice, I... | Marc Palm | 10,400 | <p>Multiplicity one refers to something else, related but much weaker.</p>
<p>For the analogue question for lattices, there are trivial counter examples: Induction by steps for example suggests on the level of Lie groups
$$ Ind_{\Gamma(N)} ^{PSL_2(\mathbb{R})} 1 \cong Ind_{PSL_2(\mathbb{Z})} ^{PSL_2(\mathbb{R})} Ind_{... |
102,624 | <p>I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one. </p>
<p>Since Maass cusp forms always are defined for a Fuchsian lattice, I... | Marc Palm | 10,400 | <p>I rethought your question and have discovered a partial answer for 1) and 2). I add this as a disjoint answer, since my other answer adresses a totally different (negative) issue.</p>
<p>In Sarnak's article <a href="http://web.math.princeton.edu/sarnak/baltimore.pdf" rel="nofollow">http://web.math.princeton.edu/sar... |
4,000,089 | <p>Let <span class="math-container">$x$</span> be an element of a Banach Algebra. Let <span class="math-container">$\lambda \in \rho(x)$</span>, where <span class="math-container">$\rho(x)$</span> is the resolvent of <span class="math-container">$x$</span>.</p>
<p>Let <span class="math-container">${d(\lambda, \sigma(x)... | Intelligenti pauca | 255,730 | <p><a href="https://math.stackexchange.com/questions/3455731/curvature-calculation-of-the-ellipse-at-the-end-of-its-axes/3457101#3457101">It is well known that</a> (see figure below, <span class="math-container">$F$</span> and <span class="math-container">$G$</span> are the foci):</p>
<p><span class="math-container">$$... |
1,569,936 | <p>Given that $a_0, a_1,...,a_{n-1} \in \mathbb{C}$ I am trying to understand how the following calculation for the determinant of the following matrix follows:
$$
\text{det}
\begin{bmatrix}
x & 0 & 0 & ... & 0 & a_0 \\
-1 & x & 0 & ... & 0 & a_1 \\
0 & -1 & x &... | Thomas | 26,188 | <p>The <a href="https://en.wikipedia.org/wiki/Laplace_expansion" rel="nofollow">determinant was expanded</a> along the first column, so you get
$$
\begin{bmatrix}
\color{red}{x} & 0 & 0 & ... & 0 & a_0 \\
\color{blue}{-1} & x & 0 & ... & 0 & a_1 \\
0 & -1 & x & ... |
4,222,110 | <p>I am following course on topology that is kind of lack luster (not made for mathematicians). The course starts off with predicate logic and axiomatic set theory (ZFC). Now, I reached a point where the author defined the partition of unity and used the set of all continuous functions between 2 sets. But at the starts... | OmG | 356,329 | <p>First, reformulate the problem as the following:</p>
<p><span class="math-container">$$
y^2 - 9x^2 = p \Rightarrow(y-3x)(y+3x)= p
$$</span></p>
<p>Now, for any given <span class="math-container">$p$</span>, find its prime factors. Then, for any 2-partitions of them, solve a simple equation system.</p>
<p>To simplify... |
1,042,375 | <p><strong>Question:</strong></p>
<blockquote>
<p>Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$.</p>
</blockquote>
<p>At first I thought this was quite obvious, but then I came up with a counterexample. What if $f(x... | Balbichi | 24,690 | <p><img src="https://i.stack.imgur.com/n6Mu0.png" alt="sign preserving property"> </p>
<p>Suppose $S$ be the set which you want to prove open.</p>
<p>$x\in S$ then $f(x)>0$ by Sign Preserving property of continuous function there is a $\delta>0$ such that $f(x)>0\forall x\in (x-\delta,x+\delta)$</p>
<p>so $... |
41,302 | <p>This is a two part question, and for that I apologize.. but they're related!</p>
<p>Here's what I'm working with:</p>
<pre><code>d1 = Import["file.CSV", "List"]
size = Length[d1]
dis1 = RandomChoice[{d1}, {100, size}]
</code></pre>
<ul>
<li><p><strong>Q1</strong>: Length views <code>d1</code> as $300,000$ indiv... | ciao | 11,467 | <p>Q1: You don't need the curly-braces around <code>d1</code> in the <code>RandomChoice</code>: That turns it into a list with one element - <code>d1</code>.</p>
<p>Q2: If memory utilization is more important than speed (because it is usually far better to generate samples/variates/etc. <em>en masse</em>), you can do ... |
59,965 | <p>If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$?</p>
<p>Thanks.</p>
| ShawnD | 14,467 | <p>If $f(x,y)=x^2+y^2$ could be written as $u(x)v(y)$, then since $f(0,0)=0$, this means either $u(0)=0$ or $v(0)=0$. If this were the case, $f(x,y)$ would be equal to zero on either the entire $y$-axis or the entire $x$-axis. This clearly is not the case, so $f(x,y)$ can not be written as $u(x)v(y)$.</p>
<p>You mig... |
908,083 | <p>I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ </p>
<p>Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does this)</p>
<p>Thanks in advance</p>
| Ross Millikan | 1,827 | <p>Yes, you are multiplying by $1$ in the form $\dfrac {\frac 12}{\frac 12}$, which takes you to $\frac {2x+1}{6x^2}$ That is all you can do</p>
|
2,311,848 | <p>$X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.</p>
<p>Here's my reasoning: </p>
<p>$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$. </p>
<p>I claim that we have 2 cases here: </p>
<p>1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\leq z)... | Graham Kemp | 135,106 | <p>Your reasoning, of using the Principle of Inclusion and Exclusion, would be fine if you were dealing with minimum rather than maximum.</p>
<p>$\bullet \quad F_{\min \{X,Y\}}(z) ~{=~ \mathsf P(X\leq z~\cup~Y\leq z) \\=~ F_X(x)+F_Y(z)-F_X(z)\cdot F_Y(z)}\\\bullet\quad F_{\max\{X,Y\}}(z)~{=~\mathsf P(X\leq z~\cap~Y\le... |
1,442,344 | <p>I would need a little help in finding the inverse Laplace transform of the function:
$$f(s)=\frac{s}{(s+1)^2(s+2)}.$$</p>
<p>Thanks in advance.</p>
| Jack D'Aurizio | 44,121 | <p><strong>Hint:</strong> recall that:
$$ \mathcal{L}^{-1}\left(\frac{1}{s+a}\right)=e^{-ax},\qquad \mathcal{L}^{-1}\left(\frac{1}{(s+a)^2}\right) = xe^{-ax} $$
and apply a partial fraction decomposition.</p>
|
1,442,344 | <p>I would need a little help in finding the inverse Laplace transform of the function:
$$f(s)=\frac{s}{(s+1)^2(s+2)}.$$</p>
<p>Thanks in advance.</p>
| Mark Watson | 272,109 | <p><strong>Hint:</strong> the partial fraction decomposition could begin with something like:</p>
<p>$$ \frac{s}{(s+1)^2(s+2)} = \frac{2s+1}{(s+1)^2} + \frac{-2}{s+2} $$</p>
<p>Now you continue from here.</p>
|
61,316 | <p>Hi all,</p>
<p>I heard a claim that if I have a matrix $A\in\mathbb R^{n\times n}$ such that $A^n \to 0 \ (\text{for }n\to\infty )$
(that is, every entry of $A^n$ converges to $0$ where $n\to \infty$)
then $I-A$ is invertible.</p>
<p>anyone knows if there is a name for such a matrix or how (for general knowledge)... | Robert Israel | 13,650 | <p>No need for an infinite series. If $I - A$ was not invertible, there would be a nonzero vector $v$ with $A v = v$, and then $A^n v = v$ for all $n$, implying $A^n$ can't go to 0 as $n \to \infty$. </p>
|
3,436,891 | <p>I have the following stupid question in my mind while i am studying for exams.
Does <span class="math-container">$X<\infty \ a.s$</span>, implies that <span class="math-container">$\mathbb E(X)<\infty$</span>? </p>
<p>Further on this, is the converse of the above statement true? Do give me a bit summary on th... | roundsquare | 706,295 | <p>The simplest counter example I can think of a random variable <span class="math-container">$X$</span> which can take values <span class="math-container">$\{1, 2, 4, ...\}$</span> where <span class="math-container">$P(X=n)=\frac{1}{2^n}$</span> if <span class="math-container">$n$</span> is a power of <span class="mat... |
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Gigili | 181,853 | <p>Prove <em>by induction</em> that:</p>
<p>$$\sum\limits_{i=1}^k i= 1 + 2+ 3 +4+...+k=\frac{k(k+1)}{2}$$</p>
<p>If it holds for $k$, it should be true for $k+1$:</p>
<p>$$\sum\limits_{i=1}^{k+1} i= \color{red}{1 + 2+ 3 +4+...+k}+(k+1)=\frac{k(k+1)}{2}+k+1=\frac{(k+1)(k+2)}{2}$$</p>
<p>The same applies for your sec... |
550,441 | <p>Say I roll a 6-sided die until its sum exceeds $X$. What is E(rolls)?</p>
| Did | 6,179 | <p>One looks for $n(x+1)$ where, for every integer $k$, $n(k)$ denotes the expected number of rolls needed to exceed $x$ starting from $x+1-k$. Thus, $n(k)=1+\frac16\sum\limits_{i=1}^6n(k-i)$ for every $k\geqslant1$ and $n(k)=0$ for every $k\leqslant0$.</p>
<p>For every $|s|\lt1$, let $N(s)=\sum\limits_{k}n(k)s^k$, th... |
3,931,246 | <p>I want to find the range of <span class="math-container">$x$</span> on which <span class="math-container">$f$</span> is decreasing, where
<span class="math-container">$$f(x)=\int_0^{x^2-x}e^{t^2-1}dt$$</span></p>
<p>Let <span class="math-container">$u=x^2-x$</span>, then <span class="math-container">$\frac{du}{dx}=2... | Fred | 380,717 | <p>Everything is fine ! A little bit more can be said:</p>
<p><span class="math-container">$f$</span> is strictly decreasing on <span class="math-container">$(-\infty,\frac{1}{2}]$</span></p>
<p>and</p>
<p><span class="math-container">$f$</span> is strictly increasing on <span class="math-container">$[\frac{1}{2}, \inf... |
4,463,559 | <p>Let <span class="math-container">$K \subset \mathbb{R}^n\times [a,b]$</span> a compact subset. For each <span class="math-container">$t \in [a,b]$</span>, let <span class="math-container">$K_t= \{x \in \mathbb{R}^n : (x,t) \in K\}$</span>. Suppose that, for all <span class="math-container">$t \in [a,b]$</span>, <spa... | André Rasera | 577,754 | <p>The use of the Lebesgue measure would simplify things a lot. But it's possible to use only the definition of the total volume of countable unions of rectangles (not necessarily open). So, let <span class="math-container">$v_p$</span> denote the total volume of a countable union of rectangles <span class="math-contai... |
65,166 | <p>For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested in the spectral gap of $G$, i.e. the first nonzero eigenvalue of $\Delta$, denoted by $\lambda_{1}(G)$.</p>
<p>Is i... | Aaron Meyerowitz | 8,008 | <p>I assume you mean to fix $d$ and let $n$ grow. When $d \gt 3$ the graph (at least in the case that all vertices have degree $r$) will likely be connected and even $d$-connected. But in case $d=2$ one has a disjoint union of cycles. Then it seems likely that for fixed $m$ there is, with probability approaching 1, a c... |
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