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 |
|---|---|---|---|---|
2,357,115 | <blockquote>
<p>$A$ is invertible matrix over $\mathbb{R}$, prove that $AA^t+A^tA$ is invertible</p>
</blockquote>
<p>It seems to be a trivial question, but it's not.
I tried using determinants, i.e $|A| \ne 0 \to |AA^t+A^tA|\ne0$, but calculating $|AA^t+A^tA|$ is not easy.</p>
| user1551 | 1,551 | <p>Both $AA^t$ and $A^tA$ are positive definite. Hence their sum is positive definite and invertible.</p>
|
2,412,454 | <p>I was obviously not clear enough in my first question, so I will reformulate. I have the following equation
$$
A=\frac{B\sin 2\theta}{C+D\cos 2\theta}
$$
where $A,B,C,D$ are variables.
I need to solve or rewrite the equation to easily obtain $\theta$ (or $2\theta$), given known values for $A, B, C, D$.
Thanks for a... | José Carlos Santos | 446,262 | <p>Yes $\emptyset$ an equivalence relation on $\emptyset$. What is $\emptyset$ as a binary relation? It's the relation such that no two elements are related.</p>
<p>However, if $A$ is not empty, then $\emptyset$ is <em>not</em> an equivalence relation on $A$ (and the authors do <em>not</em> claim that it is), because ... |
2,684,805 | <p>This question is asked by my 12 yr old cousin and I seem to be failing to give him a convincing explanation. Here is the summary of our discussion so far - </p>
<p>Case1 : $a>0, b>0$<br>
<a href="https://i.stack.imgur.com/fuoZS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fuoZS.png" alt=... | Robert Z | 299,698 | <p>If we use only two of the three colours then $(A,B,C,D)=(X,Y,X,Y)$ where $X$ can be chosen in $3$ ways and $Y$ in $2$ ways. So the number of such colourings is $3\cdot 2=6$.</p>
<p>If we use all three colours then $(A,B,C,D)=(X,Y,X,Z)$ or $(A,B,C,D)=(Y,X,Z,X)$ where $X$ can be chosen in $3$ ways and $Y$ in $2$ ways... |
110,162 | <p>One can use <code>$Epilog</code> to do something when the Kernel is quit or put an <code>end.m</code> file next to the <code>init.m</code>.</p>
<blockquote>
<p>For Wolfram System sessions, <code>$Epilog</code> is conventionally defined to read in a file named end.m.</p>
</blockquote>
<p>But if <code>$Epilog</code> i... | Chris Degnen | 363 | <p>Add the default epilog code to your own. This is what the default does.</p>
<pre><code>?? $Epilog
</code></pre>
<blockquote>
<pre><code>$Epilog:=If[FindFile[end`]=!=$Failed,<<end`]
</code></pre>
</blockquote>
|
2,781,017 | <p>I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if</p>
<p>$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$ </p>
<p>holds true for $a_i$, $b_i \in (0,1)$.</p>
| Mundron Schmidt | 448,151 | <p>Consider
$$
\sum\frac{a_i}{b_i}\leq \frac{\sum a_i}{\sum b_i} \Leftrightarrow \sum b_i\sum\frac{a_i}{b_i}\leq\sum a_i.
$$
Renaming $c_i=\frac{a_i}{b_i}>0$ implies $a_i=b_ic_i$ and
$$
\sum b_i\sum\frac{a_i}{b_i}\leq \sum a_i\Leftrightarrow \sum b_i\sum c_i\leq \sum b_ic_i.
$$</p>
|
2,948,045 | <p>In Eric Gourgoulhon's "Special Relativity in General Frames", it is claimed that the two dimensional sphere is not an affine space. Where an affine space of dimension <em>n</em> on <span class="math-container">$\mathbb R$</span> is defined to be a non-empty set E such that there exists a vector space V of dimension ... | Community | -1 | <p>Your definition is different from the definition <a href="https://en.m.wikipedia.org/wiki/Affine_space#Affine_subspaces_and_parallelism" rel="nofollow noreferrer">here</a>. (Though they are probably equivalent.)</p>
<p>Using that definition it is straight forward that <span class="math-container">$E$</span> is "fl... |
623,190 | <p>What would be the formula, to determine a rectagles edges, when given the perimeter and space? for example, the rectagles space is 80, and the perimeter is 36, and the edge would be 8 and 10, but how do I find them.</p>
<p>I know that the formula for the perimeter would be
2x+2y=per, or 2(space/y)+2y=per
However I'... | chaosflaws | 81,884 | <p>I am going to use $A$ for area and $p$ for perimeter to enhance readability.</p>
<p>$a, b > 0$ so you can divide, take roots, etc. - whatever you want. Your goal is to solve the system of equations:
$$A = a*b$$
$$p = 2a+2b$$</p>
<p>So you do the following:
$$A = a*b \implies \frac{A}{a}=b$$
$$\implies p = 2a+\f... |
312,878 | <p>Why is $\mathbb{Z} [\sqrt{24}] \ne \mathbb{Z} [\sqrt{6}]$, while $\mathbb{Q} (\sqrt{24}) = \mathbb{Q} (\sqrt{6})$ ?</p>
<p>(Just guessing, is there some implicit division operation taking $2 = \sqrt{4}$ out from under the $\sqrt{}$ which you can't do in the ring?)</p>
<p>Thanks. (I feel like I should apologize for... | Boris Novikov | 62,565 | <p>Since $\sqrt{6}\not\in\mathbb{Z} [\sqrt{24}]$.</p>
|
2,646,890 | <blockquote>
<p>If <span class="math-container">$p(x)$</span> is a polynomial of degree <span class="math-container">$n$</span> such that <span class="math-container">$$p(-2)=-15,\ p(-1)=1,\ p(0)=7,\ p(1)=9,\ p(2)=13,\ p(3)=25.$$</span>
Then smalest possible value of <span class="math-container">$n$</span> is</p>
<p>Op... | Ross Millikan | 1,827 | <p>There is exactly one fifth (or lower) degree polynomial passing through six points. You can find it, for example by Newton interpolation or by writing the polynomial as $ax^5+bx^4+\ldots+f$ and writing six simultaneous equations to relfect the data you have. Solve them for $a,b,c,d,e,f$. If $a \neq 0$ the polynom... |
1,181,269 | <p>I have a function $f(x)=x+\sin x$ and I want to prove that it is strictly increasing. A natural thing to do would be examine $f(x+\epsilon)$ for $\epsilon > 0$, and it is equal to $(x+\epsilon)+\sin(x+\epsilon)=x+\epsilon+\sin x\cos \epsilon + \sin \epsilon \cos x$.</p>
<p>Now all I need to prove is that $x+\eps... | Mark Joshi | 106,024 | <p>differentiate: you get $ 1 + \cos (x)$ this is positive except on a discrete set of points. Integrate it and you get a strictly increasing function. </p>
|
1,987,230 | <p>On Socratica, I saw a video demonstrating writing groups by writing the Cayley's table satisfying three conditions of the desired order. (1) Neutral element row and column are copies of the row and column headers. (2) Every row and column has neutral element once (3) All the elements of the set are present in each ... | Arthur | 15,500 | <p>If you are unfamiliar with modular arithmetic, then this might help: If the remainder of $\frac a5$ is $1$, that is another way of saying that there is an integer $r$ such that $a=5r+1$. Squaring both sides, we get $$a^2=(5r+1)^2=25r^2+10r+1=5(5r^2+2r)+1$$which means that the remainder of $\frac{a^2}5$ is also $1$.<... |
95,126 | <p>Consider the finite sum</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
</code></pre>
<p>Is there a way to bring <em>Mathematica</em> to calculate the limit for <code>n -> ∞</code>?</p>
<p>I have tried <code>Limit[]</code> as well as <code>NLimit[]</code> without success.</p>
| SonerAlbayrak | 49,198 | <h2>The answer</h2>
<blockquote>
<p><span class="math-container">\begin{equation}
\lim\limits_{n\rightarrow\infty}\frac{x}{n}\sum _{i=1}^n \frac{n^2}{(x (n-i)+i)^2}=\lim\limits_{a\rightarrow\infty}\left\{\begin{aligned}
a^2 \left(\pi ^2 \csc ^2(\pi a x)-\log (a)-\pi \cot (\pi a)\right)&& x>1\\
1&&a... |
3,289,401 | <p>As an example in MATLAB</p>
<pre><code>[U,S,V]=svd(randn(3,2)+1j*randn(3,2))
assert(isreal(V(1,:)))
</code></pre>
<p>Why is the first row of V purely real?</p>
| ndrizza | 147,166 | <p>The solution is to do the following projections:</p>
<p>1) Projection to unit sphere</p>
<p>2) Stereographic projection</p>
<p>3) Rescaling by radius R</p>
|
2,968,235 | <p><span class="math-container">$\log_3 4$</span> and <span class="math-container">$\log_7 10$</span>: which of these two logarithms is greater?</p>
<p>I figured out that both are between <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, then between <span class="math-container">$1$<... | Calum Gilhooley | 213,690 | <p>Here is a proof using no numbers larger than <span class="math-container">$128$</span>. We have <span class="math-container">$81 < 125 < 128$</span>, i.e. <span class="math-container">$3^4 < 5^3 < 2^7$</span>, therefore <span class="math-container">$\log_35 > \tfrac{4}{3}$</span> and <span class="math... |
1,296,230 | <p>This is from Lang's <em>Algebra</em> (page 251)</p>
<blockquote>
<p><strong>Proposition 6.11</strong> <em>Let <span class="math-container">$E/F$</span> be a normal field extension. Let <span class="math-container">$E^G$</span> be the fixed field of <span class="math-container">$\operatorname{Aut}(E/F)$</span>. Th... | Community | -1 | <p><em>Let <span class="math-container">$\operatorname{char}(k) = p > 0$</span>. Suppose that <span class="math-container">$k$</span> is perfect, that is, <span class="math-container">$k^p = k$</span>.</em></p>
<p><strong>We prove the first part of the corollary for finite extensions of <span class="math-container"... |
2,061,063 | <p>Let $X \subset C(\mathbb R;\mathbb R)$ be the space of all continuous functions $u: \mathbb R \to \mathbb R$ where </p>
<p>$$\lim_{x \to \pm \infty} u(x)=0$$</p>
<p>provided with the $\sup$-norm. Let $k \in L^1(\mathbb R)$, $u \in X$ and </p>
<p>$$(Ku)(x) := \int_\mathbb R k(x-y)u(y)\,dy, \,\,\,x \in \mathbb R.$$... | Fred | 380,717 | <p>$K$ is a bounded linear operator with</p>
<p>$||K||=\int_\mathbb R |k(s)|\, ds <1$.</p>
<p>Hence (Neumann seies !) $I-K$ is invertible.</p>
|
1,728,595 | <p>Here is the claim I'm trying to understand: Given that $N$ is an integer-valued random variable, why is it true that</p>
<p>$$Var(N) = \sum_{i=1}^\infty Var(1_{N\ge i})$$</p>
<p>For context, this is a step in the answer to exercise 4.5.10 in Rosenthal, <em>A First Look at Rigorous Probability Theory</em>, 2nd ed.,... | joriki | 6,622 | <p>This is wrong. We have <span class="math-container">$N=\sum_i1_{N\ge i}$</span>, so the equation would hold if the indicator variables were all independent. But they're not; they're all positively correlated, so we need to add their positive covariances.</p>
<p><a href="http://math.uh.edu/%7Ejosic/myweb/teaching/pro... |
162,630 | <p>Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(\mathbb{A})$-representation: $L^2(\mathbb{G}(K)\backslash \mathbb{G}(\mathbb{A}))$. It naturally contains the sub-re... | Alexander Braverman | 3,891 | <p>I think the simplest answer is this. Every reductive group $G$ comes together with a bunch of smaller reductive groups - the Levi subgroups (they are smaller in the sense that their semi-simple rank is smaller). Now, you given such a subgroup $M$ you have a way to construct representations of $G$ out of representati... |
3,063,053 | <p>I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. <br /></p>
<p><span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$</span> </p>
<p>You'll notice that using L'Hopital... | mechanodroid | 144,766 | <p>Set <span class="math-container">$x = \sinh t$</span>. We have
<span class="math-container">$$\frac{x}{\sqrt{x^2+1}}= \frac{\sinh t}{\sqrt{1+\sinh^2t}} = \frac{\sinh t}{\cosh t} = \tanh t$$</span></p>
<p><span class="math-container">$x \to \infty$</span> is equivalent to <span class="math-container">$t\to\infty$</s... |
956,235 | <p>This may be a little low-brow for this forum, but I'm trying to figure out what the common base number set is between two other sets of numbers. Here's the situation: I have received quotes from two vendors for a list of products that they sell, and the prices they have quoted are:</p>
<pre><code> Vend... | symmetricuser | 125,084 | <p>You made a minor mistake: $v(t)$ should be $112 - 32t$. This should make everything correct. As for the second part, ground means $s(t)=0$. So find the corresponding time and plug into $v(t)$ to find the impact velocity.</p>
|
748,325 | <p>In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the corresponding singular vector. Let $w$ be another linearly independent vector such that $||Aw||=\sigma$, and construc... | Etienne | 80,469 | <p>Trefethen's argument is indeed very strange.</p>
<p>As I understood the above answers, everything would be fine if one knew that $Av_1$ and $Av_1$ are orthogonal. </p>
<p>This can be done by observing that since $\sigma_1=\Vert A\Vert$ and $v_1$ is a unit vector such that $\Vert Av_1\Vert=\sigma_1$, then $v_1$ is ... |
1,075,215 | <p>Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population only has this risk factor (and no others). For any two of three factors, the probability is 0.12 tha... | megas | 191,170 | <p>First, lets write down what we know:</p>
<ul>
<li><span class="math-container">$ P(A \cap B' \cap C') = P(A' \cap B \cap C')= P(A' \cap B' \cap C) = 0.1$</span></li>
<li><span class="math-container">$P(A \cap B \cap C') = P(A \cap B' \cap C)= P(A' \cap B \cap C) = 0.12$</span></li>
<li><span class="math-container">$... |
864,237 | <p>Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a so... | Jessica B | 81,247 | <p>This set up will tell you that, essentially, $C=B/A$. Think of the first isomorphism theorem.</p>
<p>In some cases, you get that $B=A\oplus C$ (see <a href="http://en.wikipedia.org/wiki/Splitting_lemma" rel="noreferrer">Wikipedia: Splitting lemma</a>).</p>
|
3,682,661 | <p>I needed help with Part (A) without using L'Hopital's because its getting too lengthy.Can someone help me obtain solution with series without using L Hospitals rule</p>
<p>I'm trying something out with series </p>
<blockquote>
<p><a href="https://i.stack.imgur.com/FC6Xs.jpg" rel="nofollow noreferrer">Question Im... | Andrés Villa | 672,548 | <p>Note that <span class="math-container">$$\text{SU}(2):={\{A\in M(2,\mathbb{C}): \langle Ax,Ay\rangle,\; \forall x,y\in\mathbb{C}^2,\;\text{det}(A)=1}\}$$</span> where <span class="math-container">$\langle x,y\rangle:=x_1\bar{y_1}+x_2\bar{y_2}.$</span> Then <span class="math-container">$$\text{SU}(2):={\{A\in M(2,\ma... |
2,267,935 | <p>There is a fibration $SO(n-1) \mapsto SO(n) \mapsto S^{n-1}$, from basically taking the first column of the matrix in $\mathbb{R}^n$. Is this fibration trivializable? </p>
| Ted Shifrin | 71,348 | <p>$SO(n)$ is the orthonormal frame bundle of $S^{n-1}$. A trivialization of this bundle would in particular imply that $S^{n-1}$ is parallelizable (i.e., has trivial tangent bundle), so this happens precisely for $n=2$, $4$, and $8$.</p>
|
650,450 | <p>Suppose that $a_n$ and $b_n$ are Cauchy sequences, and that $a_n < b_n$ for all n. Prove that $\lim_{x \to \infty}a_n \le \lim_{x \to \infty}b_n$ for all n.</p>
<p>Is it sufficient to say that we know both Cauchy sequences must converge to the limit, and since $a_n$ is always less than $b_n$, the limits will fol... | splinter123 | 118,883 | <p>It is not true, take $b_n=1/n=-a_n$, the strict inequality is not respected at the limit.</p>
|
3,418,526 | <p>The problem is as follows:</p>
<blockquote>
<p>The figure from below shows the squared speed against distance
attained of a car. It is known that for <span class="math-container">$t=0$</span> the car is at <span class="math-container">$x=0$</span>.
Find the time which will take the car to reach <span class="m... | AgentS | 168,854 | <p>You're wrongly assuming <span class="math-container">$\color{blue}{a=1}$</span>. </p>
<p>From the kinematics equation <span class="math-container">$v^2 = 2\color{blue}{a}x+u^2$</span>, with constant acceleration,<br>
when you graph <span class="math-container">$v^2$</span> against <span class="math-container">$x$<... |
3,819,658 | <p>Calculate, <span class="math-container">$$\lim\limits_{(x,y)\to (0,0)} \dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}},$$</span> if there exist.</p>
<p>My attempt:</p>
<p>I have tried several paths, for instance: <span class="math-container">$x=0$</span>, <span class="math-container">$y=0$</span>, <span class="math-container">... | Michael Rozenberg | 190,319 | <p>Use <span class="math-container">$$0\leq \dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}}\leq|x|.$$</span></p>
|
4,330,991 | <p>I do understand that if:</p>
<p><span class="math-container">$a=b \Rightarrow a^2 = b^2 $</span></p>
<p>But clearly, the graph representing these two equations won't be the same. So, (correct me if I'm wrong) this would suggest that if you square both sides of the equation, you essentially get a different set of ans... | Peter Szilas | 408,605 | <p><span class="math-container">$1)a=b;$</span></p>
<p>Squaring yields</p>
<p><span class="math-container">$2)a^2=b^2$</span>, or <span class="math-container">$a^2-b^=0$</span>, and factoring</p>
<p><span class="math-container">$(a-b)(a+b) =0$</span>, i. e.</p>
<p><span class="math-container">$a-b=0$</span>, this is eq... |
2,919,683 | <p>I am using Monte Carlo method to evaluate the integral above:
$$\int_0^\infty \frac{x^4sin(x)}{e^{x/5}} \ dx $$
I transformed variables using $u=\frac{1}{1+x}$ so I have the following finite integral:
$$\int_0^1 \frac{(1-u)^4 sen\frac{1-u}u}{u^6e^{\frac{1-u}{5u}}} \ du $$
I wrote the following code on R:</p>
<p>set... | Henno Brandsma | 4,280 | <p>Suppose $f$ is quotient. (So for all $U \subseteq Y$, $f^{-1}[U]$ open iff $U$ is open)</p>
<p>Let's check it satisfies: $f$ is continuous and maps open saturated sets to open sets.</p>
<p>$f$ continuous is clear: if $U \subseteq Y$ is open, so is $f^{-1}[U]$, by the right to left implication in the definition of ... |
2,250,469 | <p>Let n $\geq$ 4 be an integer. Determine the number of permutations of
$\{1, 2, . . . , n\}$, in which $1$ and $2$ are next to each other, with $1$ to the left of $2$.<br>
I can't make sense of this problem statement. The way I see it, if $n$ is an integer, then the pair $1,2$ could be formed by any pair with the for... | John | 7,163 | <p>Some hints:</p>
<ol>
<li>How many places can you put $12$ in the sequence?</li>
<li>How many ways can you arrange the other $n-2$ numbers in the spaces that remain?</li>
</ol>
|
1,531,755 | <p>Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$</p>
<p>My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$
Hence by l'Hopital's rule we have that
$$\lim_{n\to \infty}{1\over -(\log{a})e^{-(\log{a})n}}={1\over -\infty}=0$$</p>
<p>Is there any other way to com... | Jimmy R. | 128,037 | <p>(Similar) By L'Hopital's Rule $$\lim_{n \to +\infty}na^n=\lim_{n \to +\infty}\frac{n}{\frac{1}{a^n}}\overset{\frac{+\infty}{+\infty}}=\lim_{n \to +\infty}\frac{1}{{-\frac{1}{a^n}\ln(a)}}=-\frac{1}{\ln(a)}\lim_{n \to +\infty} a^n =0$$</p>
|
1,531,755 | <p>Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$</p>
<p>My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$
Hence by l'Hopital's rule we have that
$$\lim_{n\to \infty}{1\over -(\log{a})e^{-(\log{a})n}}={1\over -\infty}=0$$</p>
<p>Is there any other way to com... | paw88789 | 147,810 | <p>Hint:</p>
<p>Another approach would be to look at the ratio of term $n+1$ to term $n$.</p>
<p>This gives $\frac{(n+1)a^{n+1}}{na^n}=\left(1+\frac{1}{n}\right)a$</p>
<p>Since $a<1$, this factor becomes strictly less than $1$ and then stays bounded away from $1$. Thus eventually the terms of the sequence start d... |
1,531,755 | <p>Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$</p>
<p>My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$
Hence by l'Hopital's rule we have that
$$\lim_{n\to \infty}{1\over -(\log{a})e^{-(\log{a})n}}={1\over -\infty}=0$$</p>
<p>Is there any other way to com... | lcn | 290,103 | <p>First, $a$ should be in $(0,1)$, otherwise L'Hopital's Rule won't work until the following step:</p>
<p>$\lim \limits_{n \to \infty} \frac{1}{-(\ln{a})e^{-n\ln{a}}}$</p>
<p>Second, we can use Squeeze Theorem to prove this limit.</p>
<p>Proof :</p>
<p>Let $a = \frac{1}{1+b}$, where $b>0$</p>
<p>$\because na^n... |
2,262,371 | <p>If $a,b,c$ are positive real numbers, prove that
$$\frac{2}{a+b}+\frac{2}{b+c}+ \frac{2}{c+a}≥ \frac{9}{a+b+c}$$</p>
| Dr. Sonnhard Graubner | 175,066 | <p>your inequality is equivalent to
$$2\,{a}^{3}-{a}^{2}b-{a}^{2}c-a{b}^{2}-a{c}^{2}+2\,{b}^{3}-{b}^{2}c-b{c
}^{2}+2\,{c}^{3}>0
>0$$ after Clearing the denominators and this is equivalent to
$$(a-b)(a^2-b^2)+(a-c)(a^2-c^2)+(b-c)(b^2-c^2)\geq 0$$ which is true.
the equal sign holds if $$a=b=c$$</p>
|
128,666 | <p>If we start with a number like 1234 and produce the following sum 1234 + 123 + 12 + 1 = 1370.</p>
<p>If we are given the 1370 can I retrieve the 1234? A similar question was migrated over to the Math.SE because the OP did not in any way relate it to MMa. The math given over there is in no way too tough for anyone o... | george2079 | 2,079 | <p>This is a direct implementation of this answer <a href="https://math.stackexchange.com/a/1967330/92921">https://math.stackexchange.com/a/1967330/92921</a></p>
<pre><code>m = 308460277;
Reap[NestWhile[{#[[1]] - #[[2]] Sow@Floor[Divide @@ #]],
Floor[#[[2]]/10]} &, {m,
FromDigits@ConstantArray[1, Ceil... |
3,832,684 | <p>Does the following inequality hold?
<span class="math-container">$$\sqrt {x-z} \geq \sqrt x -\sqrt{z} \ , $$</span>
for all <span class="math-container">$x \geq z \geq 0$</span>.</p>
<p>My justification
<span class="math-container">\begin{equation}
z \leq x \Rightarrow \\ \sqrt z \leq \sqrt {x} \Rightarrow \\ 2\sqr... | user | 505,767 | <p>More simply we have that</p>
<p><span class="math-container">$$x-z=\left(\sqrt x -\sqrt{z}\right)\left(\sqrt x +\sqrt{z}\right)\ge \left(\sqrt x -\sqrt{z}\right)^2$$</span></p>
|
2,174,340 | <p>Given the function $$F(X,Y,Z) = \alpha^TXYZ$$ in which $X, Y, Z $ are matrices of size $n \times n$ and $\alpha$ is a vector of size $n \times 1$, how to compute the derivative of $F$ with respect to $Y$?</p>
<p>Actually I found some related questions but did not help.</p>
<p>Edit: if the function is of the form: ... | greg | 357,854 | <p>Let ${\mathcal E}$ be the 4th-order tensor with components
$$\eqalign{
{\mathcal E}_{ijkl} &= \delta_{ik}\,\delta_{jl} \cr
}$$
Using this tensor, we can calculate the differential and gradient of the function as
$$\eqalign{
f &= a^TXYZ \cr
\cr
df &= a^T(X\,dY\,Z) \cr
&= a^T(X\,{\mathcal E}\,Z^T):... |
114,147 | <p>I have two lists let say</p>
<pre><code>listF = {{7, 2}, {2, 6}, {8, 1}, {1, 7}, {11, 8}, {6, 11}};
</code></pre>
<p>and </p>
<pre><code>newD = {{{2, 7}, {7, 9}, {9, 2}}, {{7, 2}, {2, 6}, {6, 7}}, {{7,
2}, {2, 6}, {6, 7}}, {{11, 6}, {6, 2}, {2, 11}}, {{8, 1}, {1,
7}, {7, 8}}, {{11, 1}, {1, 8}, {8, 11}}, {{1,... | Edmund | 19,542 | <p>You may use <code>OrderlessPatternSequence</code> and <code>DeleteCases</code>.</p>
<p>Build a set of <code>Alternatives</code> with <code>OrderlessPatternSequence</code> and <code>Map</code> <code>DeleteCases</code> over the sublists.</p>
<pre><code>DeleteCases[Alternatives @@ ({OrderlessPatternSequence @@ #} &am... |
634,132 | <p>Let $G$ be a cyclic group with $N$ elements. Then it follows that</p>
<p>$$N=\sum_{d|N} \sum_{g\in G,\text{ord}(g)=d} 1.$$</p>
<p>I simply can not understand this equality. I know that for every divisor $d|N$ there is a unique subgroup in $G$ of order $d$ with $\phi(d)$ elements. But how come that when you add all... | angryavian | 43,949 | <p>Hint: by definition, a function takes in some inputs, and produces a <em>unique</em> output.</p>
|
587,077 | <p>Given any prime $p$. Prove that $(p-1)! \equiv -1 \pmod p$.</p>
<p>How to prove this?</p>
| user66733 | 66,733 | <p>OK. The first thing you need to know is that if $x^2 \equiv 1 \pmod{p}$ where $p$ is a prime number then you have $x \equiv 1 \pmod{p}$ or $x\equiv -1 \pmod{p}$.</p>
<p>I'm going to state it as a Lemma and prove it:</p>
<p>Theorem: If $p$ is a prime number the only solutions to the congruence $x^2 \equiv 1 \pmod{p... |
1,063,352 | <p>$A$ and $B$ are sets and $\mathcal{F}$ is a family of sets. I'm trying to prove that</p>
<p>$\bigcap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$</p>
<p>I start with "Let $x$ be arbitrary and let $x \in \bigcap_{A \in \mathcal{F}}(B \cup A)$, which means that $\forall C \in \mathcal{F}(x \in... | Fin8ish | 166,485 | <p><strong>hint:</strong> </p>
<p>$$\displaystyle\sum_{n=0}^\infty \dfrac {5^n}{25^n + 1} = \displaystyle\sum_{n=0}^\infty \dfrac {1}{5^n + 5^{-n}}=\displaystyle\sum_{n=0}^\infty \dfrac {1}{e^{\ln(5)n} + e^{-\ln(5)n}}=\displaystyle\sum_{n=0}^\infty \dfrac {1}{2\cosh(\ln(5)n)} $$</p>
<p>if you could find a formula for... |
4,549,070 | <p>How can I prove this without using Stirling's formula?</p>
<p><span class="math-container">$${n\choose an} \le 2^{nH(a)}$$</span>
<span class="math-container">$$H(a) := -a\log_2a -(1-a)\log_2(1-a)$$</span></p>
| adrien_vdb | 1,013,037 | <p>Assuming <span class="math-container">$a\in [0,1]$</span>, you can prove it through this simple chain of (in)equalities</p>
<p><span class="math-container">$$1 = 1^n = (a+(1-a))^n = \sum_{k=0}^n \binom{n}{k}a^k(1-a)^{n-k}\geq \binom{n}{an}a^{an}(1-a)^{n-an} = \binom{n}{an}2^{-nH(a)}.$$</span></p>
<p>First step is t... |
159,529 | <p>The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?</p>
| S. Carnahan | 121 | <p>A lot has happened in the last four years, and we now have lots of positive results.</p>
<p>The current state of knowledge is given in <a href="https://arxiv.org/abs/1804.11145" rel="noreferrer">Evans-Gannon, "Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds"</a>. In particu... |
1,093,717 | <p>Let $\xi_1, \xi_2, \ldots \xi_n, \ldots$ - independent random variables having exponential distribution $p_{\xi_i} (x) = \lambda e^{- \lambda x}, \; x \ge 0$ and $p_{\xi_i} (x) = 0, \; x < 0$. Let $\nu = \min \{n \ge 1 : \xi_n > 1\}$. Need to find the distribution function of a random variable $g = \xi_1 + \xi... | ki3i | 202,257 | <p>The fact that this involves "convolutions" and sums of i.i.d. random variables makes me think of trying to deduce the distribution from Moment generating functions. Using the independence of the $\xi_i$, we have (for $t<\lambda$),</p>
<p>$$
\begin{eqnarray*}
\mathbb{E}\left[e^{t\sum\limits_{n=1}^{\nu}\xi_{n}}\ri... |
1,093,717 | <p>Let $\xi_1, \xi_2, \ldots \xi_n, \ldots$ - independent random variables having exponential distribution $p_{\xi_i} (x) = \lambda e^{- \lambda x}, \; x \ge 0$ and $p_{\xi_i} (x) = 0, \; x < 0$. Let $\nu = \min \{n \ge 1 : \xi_n > 1\}$. Need to find the distribution function of a random variable $g = \xi_1 + \xi... | Did | 6,179 | <p>There is a subtle dependence betwen $\nu$ and the sums of random variables $\xi_k$, in particular $\nu$ is not independent of $(\xi_k)$. However, note that:</p>
<ul>
<li>$g\gt1$ almost surely</li>
<li>if $\xi_1\gt1$ then $g=\xi_1$</li>
<li>if $\xi_1\lt1$ then $g=\xi_1+g'$ where $g'$ is independent of $\xi_1$ and di... |
3,282,895 | <p>How do I calculate <span class="math-container">$$\int_{0}^{2\pi} (2+4\cos(t))/(5+4\sin(t)) dt$$</span></p>
<p>I've recently started calculating integral via the residue theorem. Somehow I'm stuck with this certain integral. I've substituted t with e^it and received two polynoms but somehow I only get funny solutio... | Sohom Paul | 686,081 | <p>It looks like you are already familiar with the substitution <span class="math-container">$z=e^{it}$</span> which allows us to transform this real integral into a contour integral. Note that on the unit circle, <span class="math-container">$\bar{z} = 1/z$</span>, so we get <span class="math-container">$\cos t = (z +... |
2,933,572 | <p>Suppose <span class="math-container">$A = 1/2^{100\log(n)}$</span>, and <span class="math-container">$B = e^{-100\log(2) \log(n)}$</span>.</p>
<p>I'm required to prove that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are equal, how should I prove this? I tried applying some r... | TheSilverDoe | 594,484 | <p><span class="math-container">$C$</span> is closed </p>
<p><span class="math-container">$\Leftrightarrow$</span> <span class="math-container">$\mathbb{R} \setminus C$</span> is open </p>
<p><span class="math-container">$\Leftrightarrow$</span> <span class="math-container">$ \forall x \in \mathbb{R}\setminus C, \exi... |
4,253,160 | <p>I was recently taught that a subset W is a subspace of V if and only if:</p>
<ol>
<li>W is non-empty.</li>
<li>W is closed under vector addition.</li>
<li>W is closed under scalar multiplication.</li>
</ol>
<p>So we only need to prove 3 out of the 10 vector space axioms; why is this? Is it because it's redundant to ... | Richard Jensen | 658,583 | <p>Hint: <span class="math-container">$\cos(\frac{1}{x}) \le 1$</span>. Therefore</p>
<p><span class="math-container">$|\sin(x)\cos(1/x)|\le |\sin(x)|$</span></p>
|
192,095 | <p>Suppose I have a convex program which has only two variables, the objective function is strictly convex, and the constraints are linear functions. </p>
<p>I think removing all non-tight constraints doesn't change the optimal solution.</p>
<p>However, when there are more than 2 tight constraints, I am not sure if r... | J Fabian Meier | 3,816 | <p>In convex optimization, a local optimum is a global one. So if you do not change the neighbourhood of the optimal solution, you do not change the optimal solution.</p>
|
2,713,937 | <p>I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:</p>
<p>$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator. </p>
| Joffan | 206,402 | <p>You could prove that $1<\frac{k}{k-1} < \frac{\ell}{\ell-1}$ for $k>\ell$ and then observe that $\frac {3\times 5}{2\times4}<2$, and that choosing larger odd primes will thus make the result smaller.</p>
|
2,713,937 | <p>I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:</p>
<p>$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator. </p>
| egreg | 62,967 | <p>Assume $p>q$:
$$
2(p-1)(q-1)-pq=pq-2p-2q+2>pq-4p+2=p(q-4)+2
$$
which is $>0$ for $q>3$. For $q=3$,
$$
2(p-1)(q-1)-pq=4(p-1)-3p=p-4>0
$$
Therefore $2(p-1)(q-1)>pq$.</p>
|
2,713,937 | <p>I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:</p>
<p>$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator. </p>
| Hans | 64,809 | <p>$$\frac{pq}{(p-1)(q-1)} <2\Longleftrightarrow pq-2p-2q+2 =(p-2)(q-2)-2>0$$
for $p\ge 3$ and $q\ge 4$.</p>
|
102,427 | <p>I just coded a simple simulation module that looks at the evolution of a continuous trait in a haploid asexually reproducing population under density dependent competition in discrete time (i.e. non-overlapping generations, using recurrence equations). What I am interested in is finding out whether evolution would a... | m_goldberg | 3,066 | <p>When building a simulation like yours you should test the performance of the individual components before incorporating them into the simulation. That is, you should know the cost of the components as well as the values they return. Here is a simple example based on your code. </p>
<p>You use <code>Clip</code> in a... |
2,861,867 | <p>Let $A$ be any commutative ring (with $1$) and $x,y \in A$ such that $x+y = 1$. Then it follows that for any $k,l$, there exist $a,b \in A$ such that $ax^k+by^l = 1$. </p>
<p>(Proof: Suppose otherwise. Then, $(x^,k,y^l) \subset \mathfrak p$ for some prime ideal $\mathfrak p$. But then this implies that $x,y \in \ma... | Thomas Andrews | 7,933 | <p>Assuming $k,l$ non-negative integers.</p>
<p>Write $$1=(x+y)^{k+l}=\sum_{i=0}^{k+l}\binom{k+l}{i}x^iy^{k+l-i}$$</p>
<p>Now, for $i=0,\dots,k+l$ either $i\geq k$ or $k+l-i\geq l.$</p>
<p>So we just separate the terms. If we set:</p>
<p>$$\begin{align}a&=\sum_{i=k}^{k+l}\binom{k+l}{i}x^{i-k}y^{k+l-i}\\
b&=... |
890,313 | <p>Say the probability of an event occurring is 1/1000, and there are 1000 trials.</p>
<p>What's the expected number of events that occur? </p>
<p>I got to an answer in a quick script by doing the above 100,000 times and averaging the results. I got 0.99895, which seems like it makes sense. How would I use math to ge... | Did | 6,179 | <p>If the expected number of events occurring in each trial is 1/1000, the expected number of events occurring in 1000 trials is 1000 times 1/1000, that is, 1.</p>
|
2,344,758 | <p>Quoting from Wikipedia article on Euler's totient function theorem :---</p>
<blockquote>
<p>In general, when reducing a power of <span class="math-container">$a$</span> modulo <span class="math-container">$n$</span> (where <span class="math-container">$a$</span> and <span class="math-container">$n$</span> are coprim... | Ronald Blaak | 458,842 | <p>Euler's theorem states:</p>
<p>For all integers $a$ and $n$ with $GCD(a,n)=1$ we have $a^{\phi(n)} \equiv 1\mod n$.</p>
<p>Hence if we assume $x \geq y$ and $x \equiv y \mod \phi(n)$, there is some integer $k$ that satisfies $x = y + k \phi(n)$. Then we get:
$$
a^x = a^{y + k \phi(n)} = a^y \left[a^\phi(n)\right]^... |
367,497 | <p>Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero.
Does
$$
\lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0?
$$</p>
<p>I think the answer is no, but I can't find a counterexample.</p>
| Davide Giraudo | 9,849 | <p>Try $f_n:=\frac{\pi}2\chi_{(n,n+1)}$.</p>
|
372,064 | <p>can someone explain me why</p>
<p>$\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$</p>
<p>Many thanks</p>
| Marlo | 74,183 | <p>or use the chain rule. $\frac{1}{2}\frac{d}{dt}\big((\dot a)^2\big)=\frac{d}{dt}f(\dot a(t))$, where $f(x)=x^2$. So you first take the derivative w.r.t $f$ evaluated at $\dot a$ and then you take the time derivative of the argument of $f$, which is $\ddot a$. </p>
|
3,829,431 | <blockquote>
<p>If the area of equilateral triangle is <span class="math-container">$3\sqrt3$</span> cm<span class="math-container">$^2$</span> , then what is the height of the equilateral triangle?</p>
</blockquote>
<p>I am stuck with this question
<br>I solved it like this:
<br>Area of equilateral triangle is <span c... | SBRJCT | 39,413 | <p>Since I've accepted an answer, I thought I would post an answer that addressed my issue directly, in case someone else hits the same snag. I assumed, without checking, that the maximal ideals in <span class="math-container">$\mathbb{R}[X,Y]$</span> are correspondence with those of the form <span class="math-containe... |
213,665 | <p><strong>I've tried 3 methods but all failed to do that.</strong></p>
<p>1st Method</p>
<pre><code>Apply[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>2nd Method</p>
<pre><code>Map[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>3rd Method</p>
<pre><code>Flatten[{1, {2, {3, 4}, 5}, 6}, {2}]
</co... | Fraccalo | 40,354 | <p>This will work:</p>
<pre><code>l = {1, {2, {3, 4}, 5}, 6}
MapAt[Flatten, l, 2]
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
<p>also:</p>
<pre><code># /. x_ /; Length[x] > 1 :> Flatten@x & /@ l
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
|
2,538,297 | <p>This is my first question here so I hope I'm doing it right :) sorry otherwise!</p>
<p>As in the title, I was wondering if and when it is OK to calculate a limit i three dimensions through a substitution that "brings it down to two dimensions". Let me explain what I mean in a clearer way through an example. I was c... | Reese Johnston | 351,805 | <p>Your substitution is correct - substituting $z = xy$ isn't really "doing" anything, it's just giving a name to something. It's always fine to rename things.</p>
<p>On the other hand, replacing an expression with its limit inside a limit is <em>not</em> in general permissible. To take an excellent example: consider ... |
65,810 | <p>Recently, I have been learning about nef line bundles. I know that when $X$ is projective or Moishezon, a line bundle $L$ over $X$ is said to be nef iff $$L.C=\int_{C}c_{1}(L)\ge 0$$ for every curve $C$ in $X$.</p>
<p>Demailly gave a definition of nefness that works on an arbitrary compact complex manifold, i.e., ... | YangMills | 13,168 | <p>Yes, the equivalence is true (the second notion used to be called "metric nef" by some). This was an open problem for quite some time until it was solved in</p>
<p>M. Paun "Sur l'effectivité numérique des images inverses de fibrés en droites" Math. Ann. 310 (1998), no. 3, 411–421, see the Corollaire on page 412.</p... |
23,181 | <p>I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).
The sectors live in the complex plane, and for n even,
sector 0 and n/2 are bisected by the real axis, and the sectors are evenly spaced.</p>
<p>These branches meet at certain points, call... | Marcos Cossarini | 4,118 | <p>I don't understand how the angles of the connecting finite segments are determined, so I'll assume the angles are set so that they don't break any symmetry. First observe that the reflection wrt the real axis sends sectors 0,1,2,3,4,5 to 0,5,4,3,2,1 respectively. So in your second example, tree</p>
<p>(0,1,5)(1,2,5... |
3,174,339 | <p>Let <span class="math-container">$M$</span> be a <span class="math-container">$C^{\infty}$</span> manifold. Let <span class="math-container">$U$</span> be an open subset of <span class="math-container">$M$</span>. Now take a closed subset (with respect to the subspace topology on <span class="math-container">$U$</sp... | 5xum | 112,884 | <p><strong>Answer to old question before a significant edit was made</strong>:</p>
<p>No they do not. It's not true in euclidean space.</p>
<p>The function <span class="math-container">$$f(x)=\begin{cases}x& x\leq 0\\ x+1 & x>0\end{cases}$$</span> is an injection, however, <span class="math-container">$f((... |
3,174,339 | <p>Let <span class="math-container">$M$</span> be a <span class="math-container">$C^{\infty}$</span> manifold. Let <span class="math-container">$U$</span> be an open subset of <span class="math-container">$M$</span>. Now take a closed subset (with respect to the subspace topology on <span class="math-container">$U$</sp... | GEdgar | 442 | <p>Isn't the case <span class="math-container">$\mathbb R^n \to \mathbb R^n$</span> a result of L.E.J. Brouwer? In any case: <span class="math-container">$\mathbb R^1 \to \mathbb R^1$</span> is easy from calculus facts, while <span class="math-container">$\mathbb R^2 \to \mathbb R^2$</span> is harder, but can be prove... |
1,260,260 | <blockquote>
<p>Find, with proof, the smallest value of $N$ such that $$x^N \ge \ln x$$ for all $0 < x < \infty$. </p>
</blockquote>
<p>I thought of adding the natural logarithm to both sides and taking derivative. This gave me $N \ge \frac 1{\ln x}$. However, is there a better way to this?</p>
<p>Please note... | Apurv | 109,643 | <p>As you increase $n$, the graph of $x^n$ widens and goes away from $\ln x$. At the smallest value of $n$ (so that the inequality holds), the two graphs touch each other. So the two slopes at that point of touching must be same. $$\implies nx^{n-1}=\dfrac {1}{x}$$ $$\implies x^{n}=\dfrac {1}{n}=\ln x$$ as $x^n=\ln x$ ... |
244,333 | <p>Consider this equation : </p>
<p><span class="math-container">$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$</span></p>
<p>where <span class="math-container">$t$</span> varies from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> , and <span class="math-container... | siddhadev | 13,133 | <p>Working in polar coordinates can be very handy. As already posted by Egor Skriptunoff the differential equations would than look like this:
$$
\frac{dr}{dt} = -v - u\sin{\varphi} \\
r\frac{d\varphi}{dt} = -u\cos{\varphi}
$$
and for $\frac{dr}{dt}$ we obtain (by formal division):
$$
\frac{dr}{d\varphi} = r\frac... |
3,253,891 | <p>I'm a complete n00b at math, but I'm wondering how one would go about determining the value of <code>n</code> in the following comparison.</p>
<p><code>n * 1.5 + 12.5 = 12.5 / 2 + n</code></p>
<p>I'm new to the math StackExchange, so I'm also not sure how to properly format this question. Feel free to edit.</p>
<p><... | Vítězslav Štembera | 663,062 | <p>Let us talk here about second order partial differential equations (PDEs) with constant coefficients. The difference between elliptic PDEs and parabolic/hyperbolic PDEs is in their eigenvalues - elliptic PDEs have only complex eigenvalues. This means that they contain no real characteristics, i.e. no characteristic ... |
278,368 | <p><strong>Problem:</strong></p>
<p>Assume the number of cars passing a road crossing during an hour satisfies a Poisson distribution with parameter $\mu$, and that the number of passengers in each car satisfies a binomial distribution with parameters $n \in \mathbb{N}$ and $p \in (0,1)$. Let $Y$ denote the total numb... | eeeeeeeeee | 55,017 | <p>Calculating Var$(Y)$ using leonbloy's suggestion of the use of law of total expectation:</p>
<p>First, note that $$\text{Var}(Y|N)=\sum_{i=1}^N \text{Var}(X_i) = Nnp(1-p).$$
Now, by the <a href="http://en.wikipedia.org/wiki/Law_of_total_variance" rel="nofollow">law of total variance</a>, we have
$$\begin{align*}
\... |
2,156,357 | <p>if $H$ and $K$ are nonabelian simple groups prove that :</p>
<blockquote>
<p>$H$ $\times$ $K$ has exactly four distinct normal subgroups. </p>
</blockquote>
<p>Please help me prove this.</p>
| Andreas Caranti | 58,401 | <p>I assume for simplicity that this is an <em>internal</em> direct product, so that $H$ and $K$ are normal subgroups of the group $G = H \times K$.</p>
<p>I claim that either $L \cap H \ne 1$ or $L \cap K \ne 1$. In fact if $1 \ne (h, k) \in L$, with $h \ne 1$, since $H$ is non-abelian, there is $t \in H$ such that $... |
1,529,827 | <p>In order to make it clear, I ask three questions:</p>
<ol>
<li>Does $|2^m - 3^n|<10^6$ have any integers solution for $m>20$ ?</li>
<li>Is $ \liminf |2^m - 3^n|$ infinite ?</li>
<li>Is $ \liminf |2^m - 3^n|/m$ finite ?</li>
</ol>
| Wojowu | 127,263 | <p>The answer to question 1 is <strong>yes</strong>, since $2^{21}-3^{13}=502829$.</p>
<p>The answer to question 3 is <strong>no</strong>, and hence the answer to question 2 is <strong>yes</strong>. The following argument is adapted from Terry Tao's <a href="https://terrytao.wordpress.com/2011/08/21/hilberts-seventh-p... |
2,791,863 | <p>We need to calculating the limit
$$
\lim _{n\rightarrow \infty}((4^n+3)^{1/n}-(3^n+4)^{1/n})^{n3^n}
$$</p>
<p>I have tried taking the logarithm, but the limit doesnt seem to arrive at any familiar form.</p>
| Alex | 38,873 | <p>I think the limit should be $e^{-12}$.</p>
<p>First, after you rewrite the expression within the brackets, you get
$$
4(1+\frac{3}{4^n})^{\frac{1}{n}} - 3(1+\frac{4}{3^n})^{\frac{1}{n}}
$$
Both expressions within the two brackets can be expanded using Generalized binomial theorem, and the leading terms will be
$$... |
2,438,111 | <p>I hope my title somehow encapsulates my problem.</p>
<p>Let's say we have a 1-D Grid with the values 2,1,5,8,1,1. Imagine those values are of some physical quantity $\alpha$.
The mean of this would be $(2+1+5+8+1+1)/6 = 3$</p>
<p>Now let's say we have some function $f(x) = x^2$, which computes another quantity $\b... | Apurv Anand | 428,144 | <p>The amplitude of $A \cos x + B \sin x$ is $\sqrt{A^2+B^2}$.<br>
You can check easily by differentiating $f(x) = A \cos x + B \sin x$, which gives $\sin x = \frac{B}{\sqrt{A^2+B^2}}$ and $\cos x = \frac{A}{\sqrt{A^2+B^2}}$ and we can get the maximum value of $f(x)$.<br>
So here amplitude is 2.</p>
|
2,322,646 | <p>Let $f$ and $\varphi$ be continuous real valued functions on $\mathbb{R}$. Suppose $\varphi(x)=0$ for $|x|>5$ and that $\int_{\mathbb{R}}\varphi(x)\mathbb{d}x=1$. Show that
$$\lim_{h\to 0}\left[\frac{1}{h}\int_{\mathbb{R}}f(x-y)\varphi\left(\frac{y}{h}\right)\mathbb{d}y\right]=f(x).$$
I don't know how to proceed... | José Carlos Santos | 446,262 | <p>If you do the substitution $y=ht$ and $dy=h\,dt$, then you get$$\frac1h\int_{\mathbb R}f(x-y)\varphi\left(\frac yh\right)\,dy=\int_{\mathbb R}f(x-ht)\varphi(t)\,dt$$On the other hand,\begin{align*}\int_{\mathbb R}f(x-ht)\varphi(t)\,dt-f(x)&=\int_{\mathbb R}\bigl(f(x-ht)-f(x)\bigr)\varphi(t)\,dt\\&=\int_{-5}^... |
738,083 | <blockquote>
<p>Show that if two random variables X and Y are equal almost surely, then they
have the same distribution. Show that the reverse direction is not correct.</p>
</blockquote>
<p>If $2$ r.v are equal a.s. can we write $\mathbb P((X\in B)\triangle (Y\in B))=0$ (How to write this better ?)</p>
<p>then </... | BCLC | 140,308 | <p>Consider flipping a fair coin. Then $1_H$ and $1_T$ have the same distribution.</p>
<p>However, they are more than just not equal almost surely ($P(X \ne Y)>0$): they are almost surely not equal ($P(X \ne Y)=1$)!</p>
|
2,007,224 | <p>Analysis problem:</p>
<p><strong>Let $f$ and $g$ be differentiable on $ \mathbb R$. Suppose that $f(0)=g(0)$ and that $f' (x)$ is less or equal than $g' (x)$ for all $x$ greater or equal than $0$ Show that $f(x)$ is less or equal than$g(x)$ for all $x$ greater or equal than $0$.</strong></p>
<p>Is my proof correct... | Martin Argerami | 22,857 | <p>I don't really see how you conclude from
$
f'(c)g(b)=g'(c)f(b)
$
and $f'(c)\leq g'(c)$ that $f(b)\leq g(b)$. For instance, $0<2$ and $0\times (-1) = 2\times 0$, but you cannot conclude that $0\leq-1$. </p>
<p>As mentioned by dxiv, a proof can be achieved by using the Mean Value Theorem, applied to the function ... |
72,613 | <p>Given a list or string, how do I get a list of all (contiguous) sublists/substrings? The order is not important.</p>
<p>Example for lists:</p>
<pre><code>list = {1, 2, 3};
sublists[list]
(* {{}, {}, {}, {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}} *)
</code></pre>
<p>Example for strings:</p>
<pre><code>string =... | SquareOne | 19,960 | <p>For <strong>strings</strong>, there is also :</p>
<pre><code>string = "abcd";
StringCases[string, _ ~~ LetterCharacter ..., Overlaps -> All]
</code></pre>
<p>or rather (as @Kuba greatly suggested)</p>
<pre><code>StringCases[string, __, Overlaps -> All]
</code></pre>
<blockquote>
<p>{abcd, abc, ab, a, bc... |
432,811 | <p>I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$.</p>
<p>I know that
$$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$
but that's only valid when $\operatorname{Arg} z_1 - \operatorname{Arg} z_2 \in (-\pi,\pi]$, ... | dfeuer | 17,596 | <p>Think about the geometric significance of the difference between the arguments of two complex numbers. Then think about where in the plane $z-2$ and $z+2$ must lie to satisfy your equation.</p>
|
432,811 | <p>I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$.</p>
<p>I know that
$$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} \frac{z_1}{z_2},$$
but that's only valid when $\operatorname{Arg} z_1 - \operatorname{Arg} z_2 \in (-\pi,\pi]$, ... | Mark Bennet | 2,906 | <p>Here is a way of proceeding which depends on special features of the particular problem, so is not really general.</p>
<p>Construct an equilateral triangle on the line segment between $z-2$ and $z+2$ choosing the one in which the third vertex $V$ is nearest to the origin. Then, given the angle subtended at the orig... |
3,491,867 | <p>I'm working on an integral used to illustrate <span class="math-container">$\pi > \frac{22}{7}$</span> and I'm stuck on finding the name of a theorem for the following:</p>
<p>Let <span class="math-container">$f(x)$</span> be a continuous Real Valued function on the interval <span class="math-container">$[a,b]$<... | Stefan Perko | 166,694 | <p>A theorem stating exactly this property has (in all likelihood) no name since this is neither a deep result nor a property specific to integrals.</p>
<p>However, given an ordered real vector space <span class="math-container">$(V,\leq)$</span> and a functional (linear map) <span class="math-container">$F : V\to \ma... |
542,391 | <p>I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question:</p>
<p>Why is it that when you have an eigenvalue x=0 with algebraic multiplicity greater than 1, that you don't put a 1 in the superdiagonal ... | Marc van Leeuwen | 18,880 | <p>All Jordan <em>blocks</em> do have their entries on the super-diagonal (if any) equal to$~1$, whether the eigenvalue of the block is$~0$ or not. What confuses you is that one can have multiple Jordan blocks for the same eigenvalue; then between adjacent Jordan blocks for the same $\lambda$ there is a super-diagonal ... |
3,269,861 | <p>I have to determine a direct sum of cyclic groups to which <span class="math-container">$(\mathbb{Z}/200\mathbb{Z})^\times$</span> is isomorphic. For general <span class="math-container">$n\in\mathbb{Z}_{\geq2}$</span>, we can write <span class="math-container">$n=p_1^{e_1}\cdot\dots \cdot p_k^{e_k}$</span> for pai... | Dr. Sonnhard Graubner | 175,066 | <p>From your formula we get <span class="math-container">$$\frac{n(2a_1+(n-1)d)}{2}=n^2+3n$$</span> for <span class="math-container">$$n\neq 0$$</span> we get
<span class="math-container">$$2a_1=2n+6-(n-1)d$$</span> so...?</p>
|
3,269,861 | <p>I have to determine a direct sum of cyclic groups to which <span class="math-container">$(\mathbb{Z}/200\mathbb{Z})^\times$</span> is isomorphic. For general <span class="math-container">$n\in\mathbb{Z}_{\geq2}$</span>, we can write <span class="math-container">$n=p_1^{e_1}\cdot\dots \cdot p_k^{e_k}$</span> for pai... | lulu | 252,071 | <p>Taking <span class="math-container">$n=1$</span> we see that <span class="math-container">$S_1=a_1=4$</span>.</p>
|
4,002,458 | <p>I'm a geometry student. Recently we were doing all kinds of crazy circle stuff, and it occurred to me that I don't know why <span class="math-container">$\pi r^2$</span> is the area of a circle. I mean, how do I <em>really</em> know that's true, aside from just taking my teachers + books at their word?</p>
<p>So I t... | Ethan Bolker | 72,858 | <p>This is an excellent question. You are following in Archimedes' footsteps and starting to invent integral calculus and the idea of a limit.</p>
<p>I will try to address (briefly!) the mathematical and philosophical issues here, not the programming question.</p>
<p>You are right to worry about a process that has to g... |
363,911 | <p>If a function $f:[-2,3]\to \mathbb{R}$ is defined by </p>
<p>$f(x)=\begin{cases} 2|x|+1 \; ;\; \text{ if } x \in \Bbb Q \\ 0 \; ;\; \text{ if } x \notin \Bbb Q \end{cases}$ </p>
<p>Prove that $f$ is not Riemann integrable.</p>
<p>What I came up with:<br>
$m_k=0$,$M_k=7$ </p>
<p>Which implies $U(P,f)=35$ and ... | Did | 6,179 | <p>Let $g:[-2,3]\to\mathbb R$ and $h:[-2,3]\to\mathbb R$ be defined by $g(x)=0$ and $h(x)=2|x|+1$ for every $x$. Then every lower integral of $f$ is a lower integral of $g$ and every upper integral of $f$ is an upper integral of $h$ (can you show this?). </p>
<p>Furthermore, $g$ and $h$ are continuous hence integrable... |
821,845 | <p>As the title says, why are those two equivalent? I can find a simple derivation (using natural deduction) of $\bot$ from $\neg\neg\bot$, but i fail at proving the other implication.</p>
| Peter Smith | 35,151 | <p>(a) One of the rules of inference in standard natural deduction systems for intuitionistic logic is ex falso quodlibet, i.e.</p>
<blockquote>
<p>From $\bot$ infer $\varphi$ for any $\varphi$.</p>
</blockquote>
<p>So, as a particular application, we have a one-step derivation of, in particular, $\neg\neg\bot$ fr... |
925,140 | <p>$$f(x)=\frac { x }{ x+4 } $$</p>
<p>I am not sure how to go about solving this but here is what I have done so far:</p>
<p>$$y=\frac { x }{ x+4 } $$</p>
<p>$$(x+4)y=\frac { x }{ x+4 } (x+4)$$</p>
<p>$$yx+4y=x$$</p>
<p>I feel stuck now. Where do I go from here?</p>
| lab bhattacharjee | 33,337 | <p>$$y=f(x)=\frac x{x+4}$$</p>
<p>$$\implies f^{-1}(y)=x$$</p>
<p>Now $y=\dfrac x{x+4}$</p>
<p>Assuming $x+4\ne0, 4y+xy=x\iff x=\dfrac{4y}{1-y}$</p>
<p>Equate the values of $x$</p>
|
3,575,804 | <p>The question goes as follows. My attempts are below it. </p>
<p>A motel has ten rooms, all located on the same side of a single corridor and numbered 1 to 10 in numerical order. The motel always randomly allocates rooms to its guests. There are no other guests besides those mentioned.</p>
<p>a) Friends Molly and P... | Rezha Adrian Tanuharja | 751,970 | <p>For C:</p>
<p>The number of ways to have three rooms within 5 consecutive rooms is <span class="math-container">$3!$</span> multiplied with the number of non negative solutions of:</p>
<p><span class="math-container">$$
\begin{aligned}
A+B+C+D&=7\\
A+D&=7-i\\
C+D&=i
\end{aligned}
$$</span></p>
<p>Wher... |
3,578,191 | <p>Without tables or a calculator, find the value of <span class="math-container">$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\sqrt5}$</span>.</p>
<p>I do not understand how the positive/negative signs are obtained as shown in the book; is there a formula for expanding these kind of things (what kind of expre... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p><span class="math-container">$a-b=4$</span></p>
<p><span class="math-container">$a+b=2\sqrt5$</span></p>
<p><span class="math-container">$ab=1$</span></p>
<p><span class="math-container">$a^3-b^3=(a-b)^3+3ab(a-b)=?$</span></p>
<p><span class="math-container">$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)((a-b)^2+... |
3,691,692 | <p>Find all real values of a such that <span class="math-container">$x^2+(a+i)x-5i=0$</span> has at least one real solution. </p>
<p><span class="math-container">$$x^2+(a+i)x-5i=0$$</span></p>
<p>I have tried two ways of solving this and cannot seem to find a real solution.</p>
<p>First if I just solve for <span cla... | CopyPasteIt | 432,081 | <p>To continue from</p>
<p><span class="math-container">$\tag 1 \left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right) = \dfrac{(ad+bc)^2}{abcd}$</span></p>
<p>set </p>
<p><span class="math-container">$\quad u = ad$</span></p>
<p>and</p>
<p><span class="math-container">$\quad v = bc$</sp... |
2,256,534 | <p>As I just started learning the different rules of differentiation, I have some burning question marks in my head as such in the picture . I'm required to differentiate the following with respect to $x$.</p>
<blockquote>
<p><strong>1)</strong>
$$\frac{2x^2+4x}{x}$$
<strong>2)</strong>
$$\frac{(1-x)(x-2)}{x}$... | Sri-Amirthan Theivendran | 302,692 | <p>If you're only allowed to use sum rule, there is no real way getting around expanding the binomial. Note that
$$
f(x)
\colon=\frac{(1-x)(x-2)}{x}
=\left(1-x\right)\left(1-\frac{2}{x}\right)
=3-\frac{2}{x}-x.
$$
So
$$
f'(x)=\frac{2}{x^2}-1.
$$</p>
|
2,943,790 | <p>A function is said to be <em>continuous at zero</em> iff:</p>
<p><span class="math-container">$\lim_{x \rightarrow 0}{f(x)} = f(0)$</span></p>
<p>Could this be the same as saying:</p>
<ul>
<li>Let <span class="math-container">$\Delta$</span> = <em>the smallest open set containing zero</em></li>
<li><span class="m... | qualcuno | 362,866 | <p>As others have commented, no smaller set exist (for a standard usage of smaller, at least). In terms of open sets, what you can say is that <span class="math-container">$f$</span> is continuous at <span class="math-container">$p$</span> if for every open set <span class="math-container">$V$</span> containing <span c... |
1,652,758 | <p>the question (not homework) I am trying to answer is, in part:</p>
<blockquote>
<p><em>Let $f$ be an analytic function that maps the open unit disk $D$ into
itself and vanishes at the origin. Prove that the inequality $$|f(z)| + |f(−z)| ≤ 2 |z^2| $$ is strict, except at the origin, unless f has the
form $f(z... | Community | -1 | <p>Apply the Schwarz Lemma to
$$
g(z)=\frac{f(z)+f(-z)}{2z^2}
$$
to deduce that $|g|\le 1$. If equality holds somewhere on $|z|<1$, then $g$ is constant, by the Schwarz Lemma or the maximum principle, so
$$
f(z)+f(-z)=2\lambda z^2
$$
for some $|\lambda|=1$. This says that $f(z)=\lambda z^2 + h(z)$, with $h$ odd. We ... |
387,505 | <p>Let <span class="math-container">$f$</span> be a non-invertible bounded outer function on the unit disk. Does <span class="math-container">$f$</span> has radial limit <span class="math-container">$0$</span> somewhere? Note that such a property holds for singular inner functions.</p>
| ray | 61,993 | <p>It is worth to mention that there exists a special class of outer functions where the answer is positive: let <span class="math-container">$u$</span> be a non-constant inner function and <span class="math-container">$|\alpha|=1$</span>. Then <span class="math-container">$f:=u-\alpha$</span> is an outer function whic... |
1,821,582 | <blockquote>
<p>Find all solutions of $$\{x^3\}+[x^4]=1$$
where $[x]=\lfloor x\rfloor$</p>
</blockquote>
<p>$$$$</p>
<p>I know that $0\le\{x^3\}<1\Rightarrow 0<[x^4]\le 1$. Thus $[x^4]=1$. I couldn't get any further though since I'm having trouble with $x^4$ in the term $[x^4]$. $$$$As an example, in anoth... | Jal | 295,771 | <p>I think taking intervals for {$x$} will complicate the question. Maybe it would be easier to think whether or not {$x^3$} is zero or not. If it is not zero, the equation will have no solution.</p>
|
1,893,540 | <p>I've been asked to prove the following,
if $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$.
I tried proof by contrapositive, but I keep having trouble choosing the right $ε$. Can you guys help me out? </p>
| YoTengoUnLCD | 193,752 | <p><strong>Hint</strong>
$$
x-\varepsilon \leq y\iff x-y \leq \varepsilon
$$</p>
<p>Suppose that $x>y$. This implies that $\bar \varepsilon=\frac {x-y}{2}>0$, then $\dots$</p>
<hr>
<p>The contrapositive of $$\forall \varepsilon > 0 \, (x-\varepsilon \leq y) \rightarrow x\leq y$$
Is
$$x> y \rightarrow ... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Doug M | 317,162 | <p>I have a stupid one..</p>
<p>$\int \frac{1-y^2}{(1+y^2)^2} dy\\
\int \frac{1}{1+y^2}-\frac{2y^2}{(1+y^2)^2} dy$</p>
<p>we know that
$\int \frac{1}{1+y^2} = tan^{-1} y + c$ but what about the other term?</p>
<p>$-\int \frac{y(2y)}{(1+y^2)^2} dy\\
u = y, dv = \frac{2y}{(1+y^2)^2} dy\\
du = dy, v =-\frac{1}{(1+y^2)}... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Felix Marin | 85,343 | <p>$\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Szabolcs | 12 | <blockquote>
<h1>Status Completed</h1>
</blockquote>
<h2>Invalid attributes</h2>
<p>The Workbench has a feature where it will warn about invalid attributes in constructs similar to</p>
<pre><code>Attributes[symbol] = {attr1, attr2, ...};
</code></pre>
<p><a href="https://i.stack.imgur.com/fSuDH.png" rel="nofollow... |
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