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 |
|---|---|---|---|---|
3,321,098 | <p>I'm wondering if there is also another easy way of solving question C.
In the book, they use <span class="math-container">$A = PDP^{-1}$</span> but the only method I know is by drawing the transformation with vectors <span class="math-container">$ e_1$</span> and <span class="math-container">$e_2$</span> and then lo... | Anti | 523,511 | <p>I think I found an answer to my own question yesterday before I fell asleep ...</p>
<p>This is how the two urns look after 1 exchange of balls:</p>
<p><a href="https://i.stack.imgur.com/86akC.png" rel="nofollow noreferrer">The 2 urns after 1 exchange of balls</a></p>
<p>Now let's first define</p>
<ul>
<li><span ... |
72,794 | <p><img src="https://i.stack.imgur.com/jabWi.png" alt="enter image description here"></p>
<p>I am looking for something like this:</p>
<pre><code>SetOptions[EvaluationNotebook[], "PaperSize" -> "A3"]
</code></pre>
<p>The options are listed, but where do I find the arguments? Say for A3 paper, do I use <code>A3</c... | Jinxed | 24,763 | <p>This should get you started for A3:</p>
<pre><code>SetOptions[EvaluationNotebook[],
PrintingOptions -> {"PaperSize" ->
Round[72*
QuantityMagnitude@
UnitConvert[Quantity[#, "Millimeters"] & /@ {2*210, 297},
"Inches"]]}]
</code></pre>
<p><strong>Background</strong></p>
<p><em>Mathematica... |
3,849,649 | <p>Given <span class="math-container">$A^2=A$</span>, <span class="math-container">$2A−B−AB=I$</span>, prove that <span class="math-container">$A−B$</span> is invertible.</p>
<hr />
<p>I have got <span class="math-container">$(I+A)(A-B)$</span>, what is the next step?</p>
<p>Thanks a lot.</p>
| egreg | 62,967 | <p>It's an important fact that</p>
<blockquote>
<p>a square matrix that has a one-sided inverse is invertible</p>
</blockquote>
<p>So if you arrive at <span class="math-container">$(A+I)(A-B)=I$</span>, you're done.</p>
<p>How did you get there? From <span class="math-container">$2A-B-AB=I$</span>, you can write
<span ... |
246,827 | <p>Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite. </p>
<p>Consider, though, the results of the following numerical experiment. I generated $A$ by letting its eigenvalues be random in $[0,1]$, and selecting its eigenvectors by ge... | Guillaume Aubrun | 908 | <p>It is in theory possible to use free probability to describe the limit eigenvalue distribution (as the dimension tends to <span class="math-container">$+\infty$</span>) of the anticommutator <span class="math-container">$AB+BA$</span> when <span class="math-container">$A$</span> and <span class="math-container">$B$<... |
97,579 | <p>Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?</p>
| Suvrit | 8,430 | <p>To expand on my first comment, if $A, B > 0$ are symmetric positive definite matrices. Then, it is known that</p>
<p>$$\left(\frac{A^{-1}+B^{-1}}{2}\right)^{-1} \le A\sharp B \le \frac{A+B}{2},$$
where the inequalities are in the Löwner partial order, and $A\sharp B := A^{1/2}(A^{1/2}BA^{1/2})^{-1/2}A^{1/2}... |
3,639,122 | <blockquote>
<p>Let <span class="math-container">$X:\varOmega \to \mathbb{R}$</span> be a random variable. Random variable <span class="math-container">$X$</span> is <em>degenerate</em> if
for some <span class="math-container">$c\in \mathbb{R}$</span> we have <span class="math-container">$\mathrm{P}(X=c)=1.$</span>... | drhab | 75,923 | <p>It might be more handsome to prove the equivalence of negations:<span class="math-container">$$X\text{ is degenerated}\iff P(X<a)\in\{0,1\}\text{ for every }a\in\mathbb R$$</span></p>
<hr>
<p><span class="math-container">$\implies$</span> </p>
<p>Let it be that <span class="math-container">$P(X=c)=1$</span>. T... |
3,891,168 | <p>Let f be the function on <span class="math-container">$[-\pi,\pi]$</span> given by <span class="math-container">$f(0)=9$</span> and <span class="math-container">$f\left( x \right) = \frac{{\sin \left( {\frac{{9x}}{2}} \right)}}{{\sin \left( {\frac{x}{2}} \right)}}$</span> for <span class="math-container">$x\ne 0$</s... | Christian Blatter | 1,303 | <p>Via <span class="math-container">$$\cos t={e^{it}+e^{-it}\over2},\qquad\sin t={e^{it}-e^{-it}\over2i}$$</span>
you can easily prove that
<span class="math-container">$${\sin(9t)\over\sin t}=1+2\bigl(\cos(2t)+\cos(4t)+\cos(6t)+\cos(8t)\bigr)\ .$$</span></p>
|
649,073 | <p>Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such</p>
<p>$f(0) = 1$ and</p>
<p>$f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ </p>
<p>$\forall x \in [-1, 1]$</p>
<p>I tried putting in $x = 1$ and $x = -1$ in the second condition to find that $f(1) = f(-1) = 0$.
I also ... | Andrew Senger | 49,323 | <p>I recall solving this problem before... was it on a Putnam? Sadly, I don't remember what leap of intuition led me to the answer.</p>
<p>One may take $f(x) = \sqrt{1-x^2}$. It is clear that $f$ is continuous on $[-1,1]$ and that $f(0) = 1$, and one sees that</p>
<p>$$\frac{2-x^2}{2} \cdot \sqrt{1-\frac{x^4}{(2-x^2)... |
10,992 | <p>Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685</p>
<p>I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web ca... | user130182 | 130,182 | <p>As Mike suggested either you can input your alpha level (nominal alpha) as .10 (2*.05) to get the one tailed p value or you can divide the given probability (actual alpha) by two. Both approaches do the same thing. Remember that taking all other things constant, the probability of recting a true null is easier for a... |
2,293,934 | <p>I'm sorry for such a vague question, but I couldn't make it more plain. The other day i was working with a solution to a problem and out of curiosity i decomposed the number into its prime factors and noticed an interesting pattern. The number $216$ distribute in $2,3,2,3,2,3$ and when i grouped it, i had three pair... | scott | 330,966 | <p>To me, it's more interesting when a number has no prime factors, i.e., it is a prime number. However, being able to break a number into its prime factors can assist with mental calculations. To use your example, </p>
<blockquote>
<p>$72\times3=(9\cdot8)\cdot3=2^3\cdot3^2\cdot3=2^3\cdot3^3=6^3=216$</p>
</blockquot... |
11,651 | <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p>
<p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Ca... | Joseph O'Rourke | 511 | <p>Perhaps: The discovery <strike>a year ago</strike>
in 2015 of a new tiling of the plane by
a convex polygonal tile, found by
Mann, McLoud, and Von Derau (the latter of whom was an undergraduate at
the time of the discovery):</p>
<hr />
<img src="https://i.stack.imgur.com/eMead.png" />
<hr />
<p>Here is a nice articl... |
11,651 | <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p>
<p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Ca... | Sebastian Schoennenbeck | 7,448 | <p>Not a single answer but rather a resource:</p>
<p>The snapshots of modern mathematics from the mathematical research institute Oberwolfach (<a href="http://www.mfo.de/math-in-public/snapshots/" rel="noreferrer">http://www.mfo.de/math-in-public/snapshots/</a>) aim to provide pretty much exactly what you ask for. Res... |
2,879,041 | <blockquote>
<p>Let $f: [a,b] \to \mathbb R$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a) = 0$ and $|f'(x)|\le k|f(x)|$ for some $k$, then $f(x)$ is zero on $[a,b]$. </p>
</blockquote>
<p>I tried proving it using Legrange's Mean Value Theorem but couldn't get it.</p>
<p>$f(x)$ is differentiable ... | xbh | 514,490 | <p>As I commented, we first prove the conclusion for a smaller interval. </p>
<p><strong>Proof</strong>. $\blacktriangleleft$ We may assume $b-a > 1/(2k)$ [otherwise we proceed on $[a,b]$, and the following argument is still applicable with slight modification] First consider the interval $[a, a + 1/(2k)] \subset ... |
670,345 | <p>Here's what I came up with:</p>
<p>For this problem, I'm required to use a comparison test to determine if $\Sigma1/ln(n)^n$ converges or diverges. By intuition, I am thinking that $\Sigma1/ln(n)^n$ converges. </p>
<p>To prove that it converges by the Direct Comparison Test, I would have to find a convergent serie... | Mark Fantini | 88,052 | <p>Since you restricted your question, I'm changing my answer.</p>
<p>No. Consider $a = (1,0,\ldots,0), b = (0,1,\ldots,0)$ and $c=(0,-1,\ldots,0)$. Then $a \cdot b = a \cdot c = 0$ but $b \neq c$.</p>
<p>This is for <strong>given</strong> $a$. If this holds <strong>for all</strong> $a$ then it is true.</p>
|
1,700,457 | <p>Assume $H \leq G$ and $K \leq G$ and assume $aH=bK$ for some $a,b \in G$. Prove that $H=K$</p>
<p>would someone give me a hint. I know that definition of being a subgroup. I know that the element look like this $ah_1=bk_1$ for all $h_1 \in H$ and for all $k_1 \in K$. I do not have an idea from where should I get st... | eyedropper | 188,031 | <p>Multiply $aH=bK$ by $b^{-1}$ to get</p>
<p>$$b^{-1}aH=K$$</p>
<p>Since there is a $g\in G$ such that $g=b^{-1}a$ we get $gH=K$. This means that for some $h\in H$ and some $k\in K$ we get the equality $gh=k$. $H$ is a subgroup which means that $e\in H$. Set $h=e$ to get $g=k$ and therefore $g\in K$. $K$ is a subgro... |
2,285,752 | <p>Does $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it? </p>
<p>Should I use 'principle value' to do:
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$</p>
| Eff | 112,061 | <blockquote>
<p>Does $\int_{-\infty}^\infty \sin (t)\,\mathrm{d}t$ converge?</p>
</blockquote>
<p>No, for the limit to exist with limit $L$, then for every pair of sequences $a_n\to-\infty$ and $b_n\to\infty$, we would have that</p>
<p>$$\lim\limits_{n\to\infty} \int_{a_n}^{b_n} \sin (t)\,\mathrm{d}t = L$$
which is... |
2,371,668 | <p>For a finite group $G$ and a subgroup $H,$ Lagrange's theorem says that $|G|=|G:H||H|,$ where $|G:H|$ is the number of cosets of $H$ in $G.$<br>
My question is for any subgroup $H$ can we find another subgroup with order $|G:H|$?</p>
| DeepSea | 101,504 | <p>Alternatively using AM-GM inequality: $\sqrt{a+b} = \sqrt{1\cdot (a+b)} \le \dfrac{1+(a+b)}{2}\implies \sqrt{a+b} - \dfrac{a+b}{2} \le \dfrac{1}{2}$. Repeat this application of the AM-GM inequality for the $2$ pairs $(b,c)$, and $(c,a)$ and add up the sought inequality follows.</p>
|
640,680 | <p>Could somebody help me to solve these two unrelated questions?
I have to prove or disprove them. The first one is which I have to answer. The second one is just for me, to understand the topic better.</p>
<p>Prove or disprove the following statements:</p>
<ol>
<li><p>If the sequence $(a_n)_{n\in \mathbb{N}}$ is b... | mathemagician | 49,176 | <ol>
<li><p>is false. Counter example $a_n=(-1)^n$ the sequence alternates between 1 and -1 so is bounded, but it is not convergent. </p></li>
<li><p>is true. If the sequence converges, say to a point $L\in\mathbb{R}^d$ you have that the infinitely many terms of the tail of the sequence are in an epsilon-ball around $L... |
2,392,411 | <p>I am in Adv. Algebra 2 and I have a question. Firstly, would like to say I haven't taken algebra in a year due to geometry (stupid order they do but oh well) and I have a question understanding this: $(x+5)^{0}$. That would be $x^{0} + 5^{0}$ which then, wouldn't that be $1 + 1$ since anything that has a power of $0... | md2perpe | 168,433 | <p>Given $\epsilon>0$ you want to find $\delta>0$ such that $|(1-\frac2x)-(-1)|<\epsilon$ when $|x-1|<\delta$.</p>
<p>Now, $|(1-\frac2x)-(-1)| = |2 - \frac2x| = 2|1-\frac1x| = \frac{2}{|x|} |x-1| < \frac{2}{|x|} \delta$ when $|x-1|<\delta.$ But we need to limit $\frac{2}{|x|}.$ To do this we make sur... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| egreg | 62,967 | <p>You can certainly do it with the substitution $t=\tan(x/2)$: you have
$$
\int\frac{1}{\cos x}\,dx=
\int\frac{1+t^2}{1-t^2}\frac{2}{1+t^2}\,dt=
\int\left(\frac{1}{1+t}+\frac{1}{1-t}\right)\,dt=
\log\left|\frac{1+t}{1-t}\right|+c
$$
The back substitution gives
$$
\log\left|
\frac{\cos\frac{x}{2}+\sin\frac{x}{2}}
... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| Community | -1 | <p>When I learned how to do this, I used a much different substitution method. Sure, I'm subbing <em>something</em>, but not <span class="math-container">$tan(\frac x2)$</span>.</p>
<p><span class="math-container">$$\int\frac{1}{\cos(x)}\ dx=\int \sec(x)\ dx = \int \sec(x)\ \frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}\ dx$$... |
233,397 | <p>could anyone please clarify me the meaning of the term 'hypothesis'? </p>
<p>with relation to terms 'reasoning' and 'assumption' ?</p>
<p>Many thanks</p>
| msc zoology | 436,336 | <blockquote>
<p>Hypothesis is a kind of speculation, supposition or assumption that we make on the basis of some argument”. </p>
</blockquote>
<p>Or </p>
<blockquote>
<p>A supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation. </p>
</blockquote>
<... |
898,002 | <p>Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ exists iff $f'(c-)=g'(c+)$, where $f'$ and $g'$ are obtained using derivative rules?</p>
<p>This would seem reasonable ... | Tobias Kienzler | 163 | <p>As the other answers already mention, this won't work. Hoewever, if you are content with distributions such that $y(x)\stackrel!=y(x_0) + \int_{x_0}^x y'(\chi)\,d\chi$, then you <em>could</em> state</p>
<p>$$y'(x) = h'(c) + (g(x)-f(x))\delta(x-c)$$</p>
<p>where $h'(x)$ is $f'(x)$ or $g'(x)$ depending on the region... |
146,973 | <p>I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent.
I figured out how to do this for the minimum of $n$ variables, but i struggle with doing it for 2 with the maximum.</p>
<p>(The context in whi... | Davide Giraudo | 9,849 | <p>The sample $(X,Y)$ have a density given by $f_X(x)f_Y(y)$ since $X$ and $Y$ are independent. You have to compute
$$\iint_{\Bbb R^2}\max\{x,y\}f_X(x)f_Y(y)dxdy.$$
Cut this integral in two parts. </p>
|
4,306,776 | <p>I know a lot of similar questions have been asked here, but sadly I can't seem to wrap my head around this topic. The topic is combinatorics.</p>
<p>I am trying to get into math during high school to be more ready for college and this is so hard.</p>
<p>I have a problem.</p>
<p>Making permutation from word <strong>P... | Peter Smith | 35,151 | <p>But that initial claim [before it was edited] isn't true. You know from the upward Lowenheim-Skolem theorem that any theory with a countable model also has models of all larger cardinalities (which therefore won't be isomorphic with the countable model).</p>
<p>What <em>is</em> true is that all <em>countable</em> mo... |
4,306,776 | <p>I know a lot of similar questions have been asked here, but sadly I can't seem to wrap my head around this topic. The topic is combinatorics.</p>
<p>I am trying to get into math during high school to be more ready for college and this is so hard.</p>
<p>I have a problem.</p>
<p>Making permutation from word <strong>P... | Sahiba Arora | 266,110 | <p>The statement is proved in Theorem 2.5 <a href="https://www.math.uni-hamburg.de/home/geschke/teaching/ModelTheory.pdf" rel="nofollow noreferrer">here</a>.</p>
|
350,211 | <p>I'm looking at the following in Jech's <em>The Axiom of Choice</em> on page 20:</p>
<blockquote>
<p><strong>2.4.1.</strong> <em>Example: The Countable Axiom of Choice implies that every infinite set has a countable subset.</em></p>
<p>Proof. Let $S$ be an infinite set. Consider all finite one-to-one finite... | Cameron Buie | 28,900 | <p>We say that $S$ is infinite iff $|S|\not<|\omega|.$ Without some choice principle, we can't conclude that all infinite sets admit a countably infinite subset, as it's entirely possible that $S$ and $\omega$ are of incomparable cardinality.</p>
<p>The least choice principle sufficient to the task is: "$\omega$ is... |
4,455,201 | <p><em>Question:</em><br />
Let <span class="math-container">$\Omega=\{a,b,c\}$</span>. Give an example for <span class="math-container">$X, F_1, F_2$</span> in which
<span class="math-container">$E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$</span></p>
<p><em>My answer:</em><br />
I am not at all sure of my answer. If you hav... | X0-user-0X | 1,059,745 | <p>Here an other answer. I hope it is true.</p>
<p>So let <span class="math-container">$\Omega=\left \{ a,b,c \right \}$</span> be our probabilisable space and a r.v. <span class="math-container">$X(\omega)=1_{\omega=c}$</span>.
So obviously <span class="math-container">$\sigma(X)=\left \{ (a,b) ,(c), \Omega, \varnothi... |
162,613 | <p>How can I apply a <code>LabelingFunction</code> with an if else condition?
Suppose you have the following plot:</p>
<pre><code>BarChart[{4, 3, 2, 1}, BarOrigin -> Left, LabelingFunction ->(Placed[#,Left] &)]
</code></pre>
<p><a href="https://i.stack.imgur.com/2o9wE.png" rel="noreferrer"><img src="https:... | Szabolcs | 12 | <p>It is important to always read the "Details" section of a symbol's documentation: <a href="http://reference.wolfram.com/language/ref/LabelingFunction.html" rel="nofollow noreferrer"><code>LabelingFunction</code></a>. There is it explained how to compute the label based on the element's index rather than its value:<... |
1,751,825 | <p>Suppose there are two norms on a Banach Space: (X, $||•||_1$) and (X, $||•||_2$) where, for any sequence {$x_n$} in X, $||x_n||_1$->0 implies $||x_n||_2$->0. Show that these norms are equivalent. </p>
<p>I'm fairly certain that making an identity between (X, $||•||_1$) and (X, $||•||_2$) is a reasonable start to th... | Dalamar | 94,110 | <p>Perhaps you could work by contradiction and assume the two norms are not equivalent, that one is unbounded with respect to the other. Then you can find a sequence $(x_n)$ such that $||x_n||_1 \leq c$ (with $c$ a positive constant) and $||x_n||_2 \geq n$ $\forall n$. Now the sequence $y_n = x_n/ \sqrt n$ clearly conv... |
1,751,825 | <p>Suppose there are two norms on a Banach Space: (X, $||•||_1$) and (X, $||•||_2$) where, for any sequence {$x_n$} in X, $||x_n||_1$->0 implies $||x_n||_2$->0. Show that these norms are equivalent. </p>
<p>I'm fairly certain that making an identity between (X, $||•||_1$) and (X, $||•||_2$) is a reasonable start to th... | Rick Sanchez | 332,412 | <p>First show, using the sequence condition that $\Vert x\Vert_2 \leq c\Vert x\Vert_1$ for some $c>0$. Then, we have that the identity map $$I:X_1 \rightarrow X_2$$ is a bounded linear bijection between Banach spaces. Open mapping theorem (bounded inverse) says its inverse is also bounded...</p>
|
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | p.s. | 17,433 | <p>The other answers cover how to solve this with Lagrange multipliers, but I'd like to point out how the problem generalizes, using vector notation. If $\mathbf{x}=(1,0,4)$ and $r=\sqrt{2}$, then problem can be expressed:
$$
\min_{\|\mathbf{y}\|_2 \le r} \ \langle \mathbf{x}, \mathbf{y}\rangle = -r\|\mathbf{x}\|_{2}
... |
1,775,166 | <p>If $$T:V\rightarrow V$$ is a bijective linear transformation then can $T^2$ also be a bijective linear transformation? Can $T^n$ also be bijective linear transformation?</p>
| Patrick Abraham | 337,503 | <p>Let $T$ be a linear bijective transformation with $T:V \rightarrow V$.</p>
<p>Thus $\exists\ \hat{T} : T \circ \hat{T} = id$</p>
<p>Base: $T^n$ is bijective may n be, without loss of generality, equal to 2. $n=2$</p>
<p>$ id = T \circ id \circ \hat{T} = T \circ T \circ \hat{T} \circ \hat{T} = T^2 \circ \hat{T^2}$... |
3,114,873 | <p>Does the function <span class="math-container">$\frac{1-2xy}{x^2 +y^2}$</span> have a max or min value for <span class="math-container">$(x,y)=/=0$</span>?</p>
<p>What I've tried so far is to take the the partial derivatives:</p>
<p><span class="math-container">$$\frac{\partial f}{\partial x} = \frac{2(-x+x^2*y -... | Mark | 470,733 | <p>Look at the permutation <span class="math-container">$\sigma=(123...n)$</span>. I say <span class="math-container">$ S_{n-1},\sigma S_{n-1},\sigma^2 S_{n-1},..., \sigma^{n-1} S_{n-1}$</span> are <span class="math-container">$n$</span> distinct cosets. Can you prove it? I'll give a hint. Suppose <span class="math-con... |
1,040,846 | <p>I want to prove, that $a_n$ is a null sequence if $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1$$</p>
<p>That means that $\forall \epsilon > 0: \exists N \in \mathbb{N}: n \ge N: |\frac{a_{n+1}}{a_n} - c| < \epsilon$</p>
<p>How can I get rid of the $a_{n+1}$ and the $c$, to show $\forall \epsilon >... | lhf | 589 | <p>$$
\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1
$$
implies that $|a_{n+1}| \le b |a_{n}|$ for $n\ge N$ and $b=(c+1)/2 < 1$.
Therefore,
$|a_{n}| \le b^{n-N} |a_{N}|$ for $n\ge N$. Since $|b|<1$, we have $b^n\to 0$.</p>
|
4,403,249 | <p><span class="math-container">$$f(x,y)=(x^2+y^2)^{\left|x\right|} $$</span> find limit at <span class="math-container">$(0,0)$</span></p>
<p>moving into parametric form <span class="math-container">$f(r\cos\phi,r\sin\phi)=(r^2)^{r\left|\cos\phi\right|}= e^{2r\left|\cos\phi\right|\ln r}$</span></p>
<p><span class="mat... | A. P. | 1,027,216 | <p>We get,
<span class="math-container">\begin{align*}
\lim_{(x,y)\to (0,0)} (x^{2}+y^{2})^{|x|}&=\lim_{r\searrow 0}(r^{2})^{|r\cos \theta|}, \tag{1}\\
&=\lim_{r\searrow 0} r^{|2r\cos \theta|}, \\
&=\lim_{r\searrow 0} e^{\log r^{|2r\cos \theta|}}, \tag{2}\\
&=e^{\displaystyle \lim_{r\searrow 0} \log r^{... |
291,676 | <p>I have a question about a problem I encountered:</p>
<p>$\exists$ a,b $\epsilon$ $\mathbb{R}$+ such that $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$</p>
<p>Any tips for going about solving this?</p>
<p>I tried:</p>
<p>$\sqrt{a+b}=\sqrt{a}+\sqrt{b}$</p>
<p>$a+b=a+b$</p>
<p>I have a feeling this isn't a legal operation...</p... | Git Gud | 55,235 | <p>$\textbf{Hint:}$ Suppose such $a,b\in \mathbb{R}^+$ do exist, then square both sides of $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$.</p>
|
3,063,130 | <blockquote>
<p>Suppose function <span class="math-container">$f(z)$</span> is holomorphic on <span class="math-container">$\mathbb{D}(0,2)$</span> and <span class="math-container">$N>0$</span>
is an integer such that: <span class="math-container">$$ |f^{(N)}(0)| = N! \sup\{|f(z)|: |z|=1\} $$</span>
show that ... | Kenny Wong | 301,805 | <p>By Cauchy's formula,
<span class="math-container">$$ f^{(N)} (0) = \frac{N!}{2\pi i} \oint_{|z| = 1} \frac{f(z) \ dz}{z^{N+1}} = \frac{N!}{2\pi } \int_{0}^{2\pi} e^{-iN\theta}f(e^{i\theta}) \ d\theta.$$</span></p>
<p>Thus
<span class="math-container">$$ |f^{(N)}(0)| \leq N!\sup_{\theta \in [0, 2\pi)} |f(e^{i\theta}... |
2,076,656 | <p>As a homework, I was asked to solve this <strong>equation</strong>, $$(3x-4\lfloor x\rfloor=0),x\in \Bbb R$$ For $x\in \Bbb Z:\lfloor x\rfloor=x \implies x=0$ But for $x\not\in\Bbb Z : \lfloor x\rfloor=\frac 34x$ So now we know that $\frac 34x\in\Bbb Z$ and $x\in\Bbb R-\Bbb Z$, so maybe ? define a function such that... | Asinomás | 33,907 | <p>rewrite this as $4\lfloor x \rfloor =3x$, so that $x=\frac{n}{3}$ for some integer $n$.</p>
<p>We now have $4\lfloor\frac{n}{3}\rfloor=n$</p>
|
2,076,656 | <p>As a homework, I was asked to solve this <strong>equation</strong>, $$(3x-4\lfloor x\rfloor=0),x\in \Bbb R$$ For $x\in \Bbb Z:\lfloor x\rfloor=x \implies x=0$ But for $x\not\in\Bbb Z : \lfloor x\rfloor=\frac 34x$ So now we know that $\frac 34x\in\Bbb Z$ and $x\in\Bbb R-\Bbb Z$, so maybe ? define a function such that... | Steven Alexis Gregory | 75,410 | <h2><span class="math-container">$$3x=4\lfloor x\rfloor$$</span></h2>
<p>Suppose that, for some integer, <span class="math-container">$n$</span>, we know that <span class="math-container">$n \le x \lt n+1$</span>. Then
<span class="math-container">$\lfloor x\rfloor = n$</span> and
<span class="math-container">$$x = ... |
4,520,435 | <p>(<strong>Note:</strong> I've posted <a href="https://math.stackexchange.com/a/4539959/231608">my own answer</a>, slightly redefining trirationals to be composed of reals instead of integers and addressing the problems pointed out here. Please take note of this while reading my original question.)</p>
<p><strong>Defi... | Francis Ocoma | 231,608 | <p>If we redefine <span class="math-container">$a\unicode{x25B6}b\unicode{x25B6}c$</span> such that <span class="math-container">$a\in\Bbb{R}$</span> and <span class="math-container">$b,c\in\Bbb{R}_{>0}$</span> instead of being integers, we get the following nifty conversions between the complex polar form and the t... |
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | Spitemaster | 604,925 | <p>If it's not obvious, an examination of the question quickly reveals the trick. Say</p>
<p><span class="math-container">$$P(x)=x^n$$</span></p>
<p>Then begin long division by <span class="math-container">$Q(x)$</span>:</p>
<p><span class="math-container">$$x^n-x^n-x^{n-1}-x^{n-2}-x^{n-3}-x^{n-4}$$</span>
<span cl... |
990,418 | <p>What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. </p>
<p>I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also <a href="http://www.wolframalpha.com/input/?i=max+sin(a)%2Bsin(b)%2Bsin(c)... | Christian Blatter | 1,303 | <p>The function $f(\alpha,\beta,\gamma):=\sin\alpha+\sin\beta+\sin\gamma$ assumes a maximum on the simplex $$S:=\bigl\{(\alpha,\beta,\gamma)\>\bigm|\>\alpha\geq0, \ \beta\geq0,\ \gamma\geq0,\ \alpha+\beta+\gamma=\pi\bigr\}\ .$$
On the other hand, if $\alpha>\beta\geq0$ one has
$$\sin\alpha+\sin\beta=2\sin{\alp... |
2,692,957 | <p>I am interested in the isometries of the hyperbolic plane in the Beltrami–Klein disk model. (<a href="https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model</a>) The Wikipedia article does not say anything about the structure o... | Lee Mosher | 26,501 | <p>As you have discovered, the answers to your questions are "yes". You can read about this in <a href="http://library.msri.org/books/gt3m/" rel="nofollow noreferrer">Thurston's notes</a> which explains in detail the relations between various models of the hyperbolic plane, including a step-by-step way to get between a... |
3,082,337 | <p>Forgive my ignorance.<br>
Is the condition <span class="math-container">$x\in\mathbb{R}$</span> necessary to the set statement <span class="math-container">$\{x \in\mathbb{R} \vert x> 0\}$</span>?<br>
In other words, if <span class="math-container">$x$</span> is greater than zero, then is it not, by definition, a... | Community | -1 | <p>One more point. It might not even make sense to say <span class="math-container">$x>0$</span> unless <span class="math-container">$0$</span> and <span class="math-container">$>$</span> are defined. </p>
<p>For example the set <span class="math-container">$\{ x \in \Bbb C : x >0\}$</span> doesn't make sense... |
4,090,609 | <p>Not any even rank real vector bundle over a smooth manifold has a complex structure, because there are even dimensional manifolds that have no almost complex structures. I am curious about the following special case: Suppose <span class="math-container">$M$</span> is a real compact, connected smooth manifold and <sp... | Georges Elencwajg | 3,217 | <p>Here are some results concerning the possibility of giving the structure of a complex vector bundle to a differentiable real vector bundle <span class="math-container">$E$</span> of rank <span class="math-container">$2r$</span> on the differentiable manifold <span class="math-container">$M$</span> .<br />
The princ... |
960,026 | <p>I'm having some trouble finding out how to calculate the probability of the following problem:</p>
<p><img src="https://i.stack.imgur.com/1VdCX.png" alt="enter image description here"></p>
<p>I'm confused as to how I should be calculating the probability of B. I'm not sure how to factor in the possibility of A win... | Mark Fischler | 150,362 | <p>The answer your instructor or book is looking for is that there is a 50% chance that B wins Kentucky, and if he does not, then there is a 28% chance that he wins Tennessee. The total probability that B wins at least one is then 50% plus (1 - 50%) times 28%, for a total of 64%.</p>
<p>There are some subtleties for... |
2,685,822 | <p>How can we prove that $L = \lim_{n \to \infty}\frac{\log\left(\frac{n^n}{n!}\right)}{n} = 1$</p>
<p>This is part of a much bigger question however I have reduced my answer to this, I have to determine the limit of $\log(n^{n}/n!)/n$ when $n$ goes to infinity.</p>
<p>Apparently the answer is 1 by wolfram alpha but ... | Atmos | 516,446 | <p>Applying Stirling's formula
$$
n! \underset{(+\infty)}{\sim} \sqrt{2n \pi}\left(\frac{n}{e}\right)^n
$$
Hence
$$
\frac{n^n}{n!}\underset{(+\infty)}{\sim} \frac{n^n}{\sqrt{2n \pi}\left(\frac{n}{e}\right)^n}=\frac{e^n}{\sqrt{2n \pi}}
$$
Then
$$
\ln\left(\frac{n^n}{n!}\right)\underset{(+\infty)}{=}n\ln\left(e\right)-\f... |
718,590 | <p>Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq r\right\}$.</p>
| wckronholm | 10,449 | <p>No. If $(M,d)$ is an infinite set endowed with the discrete metric $d$, then $M$ is the closed ball around any one of its points of radius $1$. However, $M$ is not compact.</p>
<p>Given any metric space $(M,d)$, you can define a new metric space $(M,d')$ by $d'(x,y) = \min\{d(x,y),1\}$. These metric spaces have the... |
718,590 | <p>Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq r\right\}$.</p>
| Dan Rust | 29,059 | <p>Any metric space $(M,d)$ can be given a new metric which induces the same topology, but in which the entire space has finite diameter by setting $$d'(x,y)=\frac{d(x,y)}{1+d(x,y)}.$$ It follows that if $M$ is not compact, then all balls with radius larger than $1$ in this new metric $d'$ must also be non-compact.</p>... |
470,506 | <p>When is this true?
$$\lim_{r\to 0}\int_{-K}^K f(rx)dx=\int_{-K}^K \lim_{r\to 0} f(rx)dx$$
Is it true without the hypothesis of continuity of
$f$? </p>
<p>Thank you.</p>
| detnvvp | 85,818 | <p>The problem is here:</p>
<p>"As there are infinite sequences converging to points of the form $\left(\frac{1}{2^k},0\right)$, and $J$ is finite, $J$ can contain points from only a finite number of them."</p>
<p>Here $J$ is finite, but the balls you mention will contain finally all the terms of all the sequences co... |
1,046,961 | <p>Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ <br/>
I'm thinking; <br/>
Let $f(x)=-f(2x)$ <br/>
Use a substitution $x=y/2$ for $y \in \mathbb{R}$. <br/>
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$ <br/>
Im just not sure if this is a good approac... | Henry | 6,460 | <p>Hints:</p>
<ul>
<li>What is $f(0)$?</li>
<li>What does continuity when $x=0$ mean?</li>
<li>What bounds does this put on $f(x)$ for general $x$?</li>
</ul>
|
4,240,794 | <p>I was trying to obtain the square root of a matrix through the eigenvalues and eigenvectors, and there is something that doesn't add up in some of the demonstrations that I observed after getting stuck.</p>
<p>So, being <span class="math-container">$Q$</span> the eigenvector column matrix of <span class="math-contai... | Robert Lee | 695,196 | <p>You have a slight mistake in your calculations. You should have
<span class="math-container">\begin{align*}
&-2 xe^{-2x}- e^{-2x} = -2 ye^{-2y}- e^{-2y} \\
\color{blue}{\implies} &-(2x+1) e^{-2x}\cdot \frac{1}{e} = -(2y+1) e^{-2y}\cdot \frac{1}{e} \\
\color{blue}{\implies} &-(2x+1) e^{-(2x+1)} = -(2y+1) ... |
2,411,568 | <p>I would like to find all $(a,b,c)\in\mathbb{N}^3$ such that $a\times 1111+b\times 111+c\times 11=9002$.</p>
<p>It is obvious that $a+b+c \equiv 2 \pmod{10}$. I made a little computer program, that basically looped over all possiblities and I found no answer, so I just have to prove it...<br>
I am thinking of an in... | Joffan | 206,402 | <p>Working $\bmod 11$, you have:<br>
$9002\equiv 4$<br>
$11\equiv 0$<br>
$111\equiv 1$<br>
$1111\equiv 0$ </p>
<p>So we need $b\equiv 4\bmod 11$ (so assuming $a,b,c$ are non-negative, we have $b\in \{4,15,\ldots,81\}$). Any of these $b$ values produces one or more solutions; you can assume $a=0$, solve directly for $... |
2,411,568 | <p>I would like to find all $(a,b,c)\in\mathbb{N}^3$ such that $a\times 1111+b\times 111+c\times 11=9002$.</p>
<p>It is obvious that $a+b+c \equiv 2 \pmod{10}$. I made a little computer program, that basically looped over all possiblities and I found no answer, so I just have to prove it...<br>
I am thinking of an in... | Barry Cipra | 86,747 | <p>Rewrite the equation as</p>
<p>$$1000a+100(a+b)+11(a+b+c)=9002$$</p>
<p>and now look for solutions of $1000A+100B+11C=9002$ with $0\le A\le B\le C$ (or $0\lt A\lt B\lt C$ if you want <em>positive</em> integer solutions). From $11C\equiv2$ mod $100$, we see $-C\equiv99C\equiv18$ mod $100$, so $C\equiv82$ mod $100$... |
1,532,401 | <blockquote>
<p>If $X\sim \exp(\lambda)$, then $E[X\mid X>20]=20+E[X]$</p>
</blockquote>
<p>Can anyone give an intuitive proof of this property? In the context that $X$ is exponential and thereby memoryless. I am having trouble in visualizing continuous memoryless distributions. Any intuitive example for the sam... | Ross Millikan | 1,827 | <p>The intuitive answer is that memoryless means that if you are waiting for an event that has not yet happened, it doesn't matter how long you have been waiting. If you get to $x=20$ and it hasn't happened, the expected time until it happens is the same from now as it was at the start. Since you have already waited ... |
1,532,401 | <blockquote>
<p>If $X\sim \exp(\lambda)$, then $E[X\mid X>20]=20+E[X]$</p>
</blockquote>
<p>Can anyone give an intuitive proof of this property? In the context that $X$ is exponential and thereby memoryless. I am having trouble in visualizing continuous memoryless distributions. Any intuitive example for the sam... | Michael Hardy | 11,667 | <p>Memorylessness says the conditional distribution of $X-20$ given $X>20$ is the same as the distribution of $X$.</p>
<p>Therefore the conditional expected value of $X-20$ given $X>20$ is the same as the expected value of $X$:</p>
<p>$$
\operatorname{E} (X - 20 \mid X>20) = \operatorname{E}(X).
$$</p>
<p>B... |
248,658 | <p>Based on the next relation:</p>
<p>$$\det\begin{bmatrix}A & B \\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B),$$</p>
<p>I have that for computing the eigenvalues of the block matrix:</p>
<p>$$\det\begin{bmatrix}A-\lambda I & B \\ C & D-\lambda I\end{bmatrix} = \det(A-\lambda I)\det((D-\lambda I)... | Hans Engler | 9,787 | <p>If $\det(A - \lambda I) = 0$, then you cannot form $(A - \lambda I)^{-1}$, which appears also in the formula. Hence the formula does not apply, unless $B = 0$ or $C = 0$, the case of a block tridiagonal matrix.</p>
|
2,907,378 | <p>In my Math book I'm solving a case where this is the situation:</p>
<p>"The demand curve for good X is linear. At a price (p) of 300 the demand is 600 units. At a price of 680 the demand is 220 units. Also the supply curve for good X is linear. If the price is 400 then the supply equals 200 units, whereas for a pri... | David K | 139,123 | <p>For supply you have computed the following:</p>
<blockquote>
<p><code>SLOPEqs = delta y / delta x = 0.5</code> (<code>delta y 800-400, delta x 1000-200</code>)</p>
<p>If I try to verify this for the first supply point (800, 1000) I would get:</p>
<p><code>Supply(x) = 0,5x + 600</code></p>
</blockquote>
<p>Presumably... |
65,220 | <p>Si $f$ es una función continua y si $f(m/2^n)=0$ para todo entero $m$ y todo natural $n$, ¿cómo demuestro que $f(x)=0$ para todo numero real $x$?</p>
<hr/>
<p><em>[translation by mixedmath]</em></p>
<p>If $f$ is a continuous function and if $f\left(\dfrac{m}{2^n}\right) = 0$ for all integers $m$ and... | Brian M. Scott | 12,042 | <p>Let $D = \left\{\frac{m}{2^n}:m\in\mathbb{Z} \land n\in\mathbb{N}\right\}$. (Members of $D$ are called <em>dyadic rationals</em>.) Since $f$ is continuous, you know that $\lim\limits_{n\to\infty}f(x_n)=f(x)$ whenever $\lim\limits_{n\to\infty}x_n=x$. Let $x$ be any real number. If we can find a sequence $\langle x_n:... |
65,220 | <p>Si $f$ es una función continua y si $f(m/2^n)=0$ para todo entero $m$ y todo natural $n$, ¿cómo demuestro que $f(x)=0$ para todo numero real $x$?</p>
<hr/>
<p><em>[translation by mixedmath]</em></p>
<p>If $f$ is a continuous function and if $f\left(\dfrac{m}{2^n}\right) = 0$ for all integers $m$ and... | André Nicolas | 6,312 | <p><strong>Added</strong>: The question changed while the answer below was being typed. A short note is added at the end to answer the altered question.</p>
<p>The wording of the original question does not accurately convey the intention. I believe that the OP <em>wants</em> a proof that every real number $x$ is the... |
41,450 | <p>I want to perform a Pearson's $\chi^2$ test to analyse contingency tables; but because I have small numbers, it is recommended to perform instead what is called a Fisher's Exact Test.</p>
<p>This requires generating all integer matrices with the same column and row totals as the one given, and compute and sum all p... | Romke Bontekoe | 1,178 | <p>Maybe you are willing to consider a Bayesian approach to this perennial problem. Beware though: Bayesians have no random variables, no p-values, no null hypotheses, etc. They have probabilities, or ratios thereof.</p>
<p>The (out of print) book "Rational Descriptions, Decisions and Designs" by Miron Tribus (1969!) ... |
89,987 | <p>In the sequence: </p>
<p>$$\lim _{n\rightarrow \infty }{\frac {n+1}{2\,n+3}}\neq 3$$</p>
<p>I know how to prove that the limit is actually $1/2$, but is there another way to prove that 1 is NOT the limit? </p>
<p>I tried to prove by negation showing that if 1 is the limit I can't find an $N$ that for every epsilo... | Will Jagy | 10,400 | <p>Yes, the value of the fraction is always below 1/2. So it stays far away from 3, indeed never gets within 5/2 of it. Note that it is not necessary to know that there is any limit to do answer your question. Anyway, take $\varepsilon = 2$ and no $n$ ever gets you within $\varepsilon$ of 3.</p>
|
4,610,313 | <p>I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:</p>
<blockquote>
<p>Given that <span class="math-container">$\log_{4n} 40\sqrt{3} = \log_{3n} 45$</span>, find <span class="math-container">$n^3$</span> (MA<span class="math-container">$\Theta$</span> 1991).</p>
</blockquot... | lone student | 460,967 | <p>I would go the following way.</p>
<p>You have :</p>
<p><span class="math-container">$$\begin{align}&\log_{4n} 40\sqrt{3} = \log_{3n} 45\\
\implies &\log_{4n} \left(3\cdot 40^2\right) = \log_{3n} 45^2=k
\end{align}$$</span></p>
<p>This leads to :</p>
<p><span class="math-container">$$\begin{align}&\begin{... |
265,801 | <p>Consider a 12-sided fair die. What is the distribution of the number T of rolls required to roll a 1, a 2, a 3, and a 4?</p>
<p>Taking inspiration from the Coupon Collector's Problem, I believe that the expected number of rolls to achieve the goal would be</p>
<p>$$E[T] = \sum\limits_{i=0}^3 \frac{12}{4-i} = 25$$... | Henry | 6,460 | <p>Personally I would call $P(n,b,k,j)$ the probability that after rolling $n$ dice with $b$ sides, $j$ of the target $k$ sides had been found, with the formula</p>
<p>$$P(n+1,b,k,j)= \frac{b-k+j}{b} P(n,b,k,j) + \frac{k+1-j}{b} P(n,b,k,j-1)$$</p>
<p>starting from $P(0,b,k,j)=0$ for $j \not = 0$ and $P(0,b,k,0)=1... |
265,801 | <p>Consider a 12-sided fair die. What is the distribution of the number T of rolls required to roll a 1, a 2, a 3, and a 4?</p>
<p>Taking inspiration from the Coupon Collector's Problem, I believe that the expected number of rolls to achieve the goal would be</p>
<p>$$E[T] = \sum\limits_{i=0}^3 \frac{12}{4-i} = 25$$... | joriki | 6,622 | <p>You can get this distribution by conditioning on the number $N$ of rolls up to $4$ required:</p>
<p>\begin{align}
\textsf{Pr}(T\le t)
&=\left(\frac23\right)^t\sum_{n=0}^t\binom tn2^{-n}\textsf{Pr}(N\le n)
\\
&=\left(\frac23\right)^t\sum_{n=0}^t\binom tn2^{-n}\frac{4!}{4^n}\left\{n\atop4\right\}
\\
&=\le... |
1,288,461 | <p>Here's the proof I was given:</p>
<p>Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense.</p>
<p>Proof:
Suppose otherwise, then there exists real numbers $a$ and $b$, with $a < b$, such that there is no $a < x < b$ with $x \in \mathbb{R} \setminus A $. To put it in another way, if $a... | hamid kamali | 208,328 | <p>$2a_n-a_{n-1}=4$. From here: $2a_nx^n-a_{n-1}x^n=4x^n$. By summation:
$$
\begin{align}
\sum_{n=1}^\infty (2a_nx^n-a_{n-1}x^n)&=\sum_{n=1}^\infty 4x^n\\
&=\frac{4x}{1-x}\,\,\,\, for\,\, |x|\lt 1
\end{align}
$$ For the left side of above equation, we can write:
$$
\begin{align}
\sum_{n=1}^\infty (2a_nx^n-a_{n... |
268,308 | <p>In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic? </p>
| Calvin Lin | 54,563 | <p>With all proofs (in whatever format), you must first identify your intended audience. That will allow you to decide what you can / cannot assume, and how to include in the right amount of detail.</p>
<p>Are they students in your field that are interested in understanding the buildup of the theory? Are they 'profess... |
2,072,866 | <p>My notes say that if $F$ is conservative (i.e. $F=\nabla f$) then $\text{curl }(F)=0$. But I feel this is not quite right.</p>
<p>There is a theorem that says that if $f(x,y,z)$ has continuous second order partial derivatives then $\text{curl }(\nabla f)=0$. (I'm fine with this theorem, it makes sense.) </p>
<p>O... | nullUser | 17,459 | <p>The proof is essentially $\partial_{ij}f = \partial_{ji}f$, and the only hypotheses needed are "whenever this holds". Recall that the curl of $F$ is zero iff $\partial_i F_j = \partial_j F_i$ for all $i,j$. When $F$ is conservative, we have that $\partial_i F_j = \partial_i \partial_j f$ and $\partial_j F_i = \parti... |
3,011,216 | <p>If <span class="math-container">$F(x)$</span> is a continuous function in <span class="math-container">$[0,1]$</span> and <span class="math-container">$F(x) = 1$</span> for all rational numbers then <span class="math-container">$F(1/\sqrt2) = 1$</span>. (True/ False)</p>
<p>I think the statement is true because sin... | mlerma54 | 406,626 | <p>The statement is true, although I would express it: there is a sequence <span class="math-container">$\{r_n\}_{n\in\mathbb{N}}$</span> of rational numbers in the interval <span class="math-container">$[0,1]$</span> such that <span class="math-container">$r_n \to 1/\sqrt{2}$</span>, hence by continuity
<span class="m... |
2,798,338 | <p>Let $G$ be a group of order 20, Prove that $G$ has a normal subgroup of order 5.</p>
<p>Obviously by Sylow theorem there is a subgroup of order 5, and since all Sylow p-subgroups are conjugate the only problem is to show that there is only one sungroup of order 5.</p>
<p>any help would be appreciated</p>
| thesmallprint | 438,651 | <p>$|G|=20=5\cdot 2^2$. Now, let $n_p(G)$ be the number of Sylow $p$-subgroups of $G$.</p>
<p>Then, Sylow III says</p>
<ul>
<li>$n_5(G)\equiv 1\bmod 5$</li>
<li>$n_5(G)$ divides $4$</li>
</ul>
<p>So, $n_5(G)=1$. This means there is only one Sylow $5$-subgroup of $G$; which in turn, is a normal subgroup of $G$.</p>
|
1,915,273 | <p>$$\int_0^r\sqrt{x-x^2}.dx$$</p>
<p>I only have basic calculus and would like to know how would one go about integrating an expression of this form.
I have tried substituting say $u=x-x^2$ but I'm still left with an $x$.</p>
<p>The method is not in my book and I can't find a similar example anywhere.</p>
<p>A hint... | Jan Eerland | 226,665 | <p>$$\mathcal{I}(\text{r})=\int_0^\text{r}\sqrt{x-x^2}\space\text{d}x=\int_0^\text{r}\sqrt{\frac{1}{4}-\left(x-\frac{1}{2}\right)^2}\space\text{d}x=$$</p>
<hr>
<p>Substitute $u=x-\frac{1}{2}$ and $\text{d}u=\text{d}x$.</p>
<p>This gives a new lower bound $u=0-\frac{1}{2}=-\frac{1}{2}$ and upper bound $u=\text{r}-\fr... |
1,915,273 | <p>$$\int_0^r\sqrt{x-x^2}.dx$$</p>
<p>I only have basic calculus and would like to know how would one go about integrating an expression of this form.
I have tried substituting say $u=x-x^2$ but I'm still left with an $x$.</p>
<p>The method is not in my book and I can't find a similar example anywhere.</p>
<p>A hint... | user84413 | 84,413 | <p>Let $\sqrt{x}=\sin t$, so $x=\sin^2 t,\; dx=2\sin t\cos t \,dt$.</p>
<p>Then $\displaystyle\int\sqrt{x-x^2}dx=\int(\sin t\cos t)(2\sin t\cos t)dt=\int2\sin^2 t\cos^2t dt=\frac{1}{2}\int\sin^2 2t dt$</p>
<p>$\;\;\;=\displaystyle\frac{1}{4}\int(1-\cos 4t)dt=\frac{1}{4}\left[t-\frac{1}{4}\sin 4t\right]+C=\frac{1}{4}\... |
3,636,563 | <p>How to find the values of <span class="math-container">$x,y$</span> and <span class="math-container">$z$</span> if
<span class="math-container">$3x²-3(1126)x=96y²+24(124)y=8z²-4(734)z $</span>?</p>
<p>I dont have any idea!! I think we can have many values of <span class="math-container">$x,y$</span> and <span clas... | Aditya Sriram | 636,845 | <p>Let us assume that the antiderivative of <span class="math-container">$f$</span> is <span class="math-container">$F$</span>.
By the Fundamental Theorem of Calculus,
<span class="math-container">$$\therefore \int_{\frac{1}{k}}^{\frac{1}{k-1}}f(x)dx =F(\frac{1}{k-1})-F(\frac{1}{k})$$</span>
<span class="math-container... |
3,603,066 | <p>My question: How does one distinguish between two embeddings of the same graph on the plane? </p>
<p>For instance, are two such embeddings considered the same if the degree sequence of their faces are the same? Or does even the slightest change, such as drawing an edge more crooked in one embedding than the other, ... | Allawonder | 145,126 | <p>Let the rate per annum be <span class="math-container">$r.$</span> Then we have that after the first year, the height of the tree is <span class="math-container">$3(1+r).$</span> After the second year that's <span class="math-container">$3(1+r)^2,$</span> and after the twelfth year the tree is <span class="math-cont... |
6,155 | <p>I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and write $0$ for the initial object), and $\otimes$ distributes over $\oplus$.</p>
<p>By an <em>exponential monad</em>, ... | Michael Ortiz | 1,938 | <p>There is a nice discussion about multiplicative sequences, &c., in Lawson and Michelsohn's book "Spin Geometry". It discusses things like the Todd genus, the A-hat genus, and so on, but also the Chern character and the ring homomorphism from K-theory to ordinary cohomology. It is a readable exposition and perhap... |
181,327 | <p>If $q$ and $w$ are the roots of the equation $$2x^2-px+7=0$$ </p>
<p>Then $q/w$ is a root of ? </p>
<p>P.s:- It is an another question of <a href="https://math.stackexchange.com/questions/181305/how-do-i-transform-the-equation-based-on-this-condition">How do I transform the equation based on this condition?</a></p... | Emily | 31,475 | <p>The methods to be used will be highly dependent on the character of $f$. If it is non-convex, there are many such algorithms; see <a href="https://mathoverflow.net/questions/32533/is-all-non-convex-optimization-heuristic">this MO post</a> for instance. </p>
<p>You might also wish to look into evolutionary algorithm... |
2,021,734 | <p>Let $p$ be a prime number and $\mu_{p^{\infty}}$ denote $$\{ z\in \mathbb{C} : \exists k \ge 1 : z^{p^k}=1 \}$$ find all of its subgroups.</p>
<p>I was able to prove that its finite subgroups are of the form $\mu_{p^n}$, where $n$ is a positive integer (using Lagrange's Theorem and the fact that in, $\mu_{\infty}=\... | Couchy | 87,768 | <p>To expand on Andreas Caranti's answer, consider a sequence $a_1,a_2,\ldots\in\mathbb N$ so that we have the subgroup </p>
<p>$$H=\bigcup_{k=1}^\infty\mu_{p^{a_k}},$$
(where any subgroup $H\leq \mu_{p^\infty}$ can be written in such a way) then since for any $k\leq a_j$ we have $\mu_{p^{k}}\leq\mu_{p^{a_j}}$, then f... |
179,322 | <p>I'm interested in classes C of $R^1$-valued random variables which possess the following properties:</p>
<p>1) the sum of two independent random variables from class C belongs to class C;</p>
<p>2) for any $\lambda \in R^1$, $\xi \in C$ we have $\lambda\xi \in C$; </p>
<p>3) any random variable from class $C$ has... | Donu Arapura | 4,144 | <p>Let me supplement Vivek's nice answer with a few additional comments. People coming from differential geometry typically normalize the $n$th Chern class -- in the Chern-Weil formula -- by a factor of $1/(2\pi i)^n$ to get it to be integral. However, if you want compatibility with algebraic definitions, then you don'... |
1,119,421 | <p>We know that p → q is not equivalent to q → p. But suppose we make a proof system that has all the rules of logical identities plus the rule (“commutativity of implies”) p → q ≃ q → p. (We are using the new symbol ≃ because these are not really equivalent.) </p>
<p>This new proof system is not sound. The point of t... | ajotatxe | 132,456 | <p>Since $\Bbb Z_6$ is cyclic and generated by $1$, the value $f(1)$ determines the homomorphism. On the other hand, $f(1)$ must have order $6$ or a divisor of $6$, so $f(1)\in\{0,3,6,9,12,15\}$. This yields six homomorphisms.</p>
|
80,432 | <p>I know Mathematica's if format is </p>
<pre><code>If[test, then result, else alternative]
</code></pre>
<p>For example, this</p>
<pre><code>y:=If[RandomReal[]<0.2, 1, 3.14]
</code></pre>
<p>would take a random real number between $0$ and $1$, and evaluate it. If it's less than $0.2$, it'll map <code>y</code> ... | Mr.Wizard | 121 | <p>There are several functions and methods available which different strengths and limitations that can guide your choice. Among them:</p>
<h2><a href="http://reference.wolfram.com/language/ref/Which.html" rel="noreferrer"><code>Which</code></a></h2>
<p>Here combined with <a href="http://reference.wolfram.com/langua... |
3,490,198 | <p>For all <span class="math-container">$a,b \in \mathbb{Z}$</span> and for all <span class="math-container">$m,n \in \mathbb{N}\setminus \left\lbrace0\right\rbrace$</span>,</p>
<p>is <span class="math-container">$a^{48m+1}+b^{48n+1} \equiv 0 \pmod{39} \iff a+b \equiv 0 \pmod{39}$</span>?</p>
<p>I think the answer is... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>Using <a href="http://mathworld.wolfram.com/FermatsLittleTheorem.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/FermatsLittleTheorem.html</a></p>
<p>For <span class="math-container">$3\mid x(x^2-1),$</span></p>
<p>and <span class="math-container">$13|x(x^{12}-1)$</span></p>
<p><span cla... |
1,285,870 | <p>Is the collection of all cardinalities a set or a proper class?
Does anybody ever think about the problem?</p>
| Trevor Wilson | 39,378 | <p>The collection of all cardinal numbers is a proper class. If it were a set $X$, then $(\sup X)^+$ would be a cardinal number greater than every element of $X$, which is a contradiction.</p>
<p>(The supremum of a set of cardinals is given by its union, and the cardinal successor $\kappa^+$ of a cardinal $\kappa$ is... |
1,368,180 | <p>Here is a tricky compass and straightedge construction problem. </p>
<blockquote>
<p>Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such that $\angle APB = \angle BAC$. </p>
</blockquote>
<p>This configuration appears frequently in O... | math_lover | 189,736 | <p>Construct isosceles triangle QAB with QA equals QB and angle QAB equals 90- BAC/2. The circumcircle of QAB is the locus of points X ( above the line AB) such that angle AXB equals BAC.. The desired point P is the intersection of this circle with CD.</p>
|
760,739 | <p>Find the limit of the sequence given by
$$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$</p>
<p>I think the answer is $\frac{20}{7}$ after dividing, but is that right? </p>
| user2345215 | 131,872 | <p>Yes it is, because
$$\frac{20n^4+12n+10}{7n^4 + 5n^3 - 20}=\frac{20+\frac{12}{n^3}+\frac{10}{n^4}}{7+\frac5n-\frac{20}{n^4}}\to\frac{20+0+0}{7+0-0}=\frac{20}7$$
by the arithmetic of limits.</p>
<p>You should see this method works in general for a fraction of two polynomials, just divide $n^d$, where $d$ is the degr... |
1,681,872 | <p>My task is this:</p>
<p>Use cylinder coordinates to calculate:$$\iiint\limits_{A}z\sqrt{x^2 + y^2}dA, \enspace A = \left\{(x,y,z):x^2 + (y - 1)^2 \leq 1,\: 0 \leq z \leq 2\right\}.$$</p>
<p>My works so far is this;</p>
<p>Switching to cylindrical coordinates we get:$$A = \left\{(r,\theta,z):0\leq r\leq 1,\: 0\leq... | cpiegore | 268,070 | <p>It's actually easier to evaluate the integral as it stands. I entered</p>
<p>$\int_{0}^{2}\int_{0}^{2}\int_{-\sqrt{2y-y^2}}^{\sqrt{2y-y^2}}z\sqrt{x^2+y^2} dxdydz$</p>
<p>in my calculator. It returned $64/9.$ You may want to look at <a href="https://math.stackexchange.com/questions/1677123/integrate-iint-limits-... |
1,681,872 | <p>My task is this:</p>
<p>Use cylinder coordinates to calculate:$$\iiint\limits_{A}z\sqrt{x^2 + y^2}dA, \enspace A = \left\{(x,y,z):x^2 + (y - 1)^2 \leq 1,\: 0 \leq z \leq 2\right\}.$$</p>
<p>My works so far is this;</p>
<p>Switching to cylindrical coordinates we get:$$A = \left\{(r,\theta,z):0\leq r\leq 1,\: 0\leq... | Kuifje | 273,220 | <p>Your can use a simpler (?) transformation with cylindrical coordinates. Given that <span class="math-container">$x^2+(y-1)^2=1$</span> is a cylinder shifted 1 unit along the <span class="math-container">$y$</span> axis, you should not consider points in the <span class="math-container">$y<0$</span> plane (<span c... |
698,460 | <p>I've literally tried everything including proofs by cases. If I were to try a an odd integer, then I would get $8k^3 + 4k^2 + 8k^2 + 4k + 1$, which obviously couldn't convert to a $\mod 7$, and I'm honestly not sure what I can do! Does anyone here have any tips?</p>
<p>Note, this is a soft question.</p>
| NasuSama | 67,036 | <p>The following might help you out when determining the integrals without complete computation.</p>
<p><strong>Hint</strong>: Think $e^{-inx}$ of Euler's identity. We get $\cos(nx) - i\sin(nx)$. Now, look at the integral carefully. If we multiply each term by term and then rewrite the whole integral as two integra... |
718,664 | <p>I'm trying to find a small primitive root modulo $p^k$, where $p$ is prime. My strategy is to test small numbers $g=2,3,\ldots$ until I find a primitive root modulo $p$. That is, until $ord_pg=\phi(p)=p-1$. There are results that suggest such a search won't take long.</p>
<p>To go from modulo $p$ to modulo $p^k$... | Álvaro Lozano-Robledo | 14,699 | <p>For your first question, consider $p=71$ and $g=11$. Here, $11$ is a primitive root modulo $71$, but $11^{70}\equiv 1 \bmod 71^2$. Unfortunately, this is not an example of your second question, because $7\bmod 71$ is also a primitive root.</p>
<p>I found this example looking at lists of <a href="https://oeis.org/A1... |
4,589,828 | <p>I'm stuck in proving the function <span class="math-container">$$f(x) = \vert x + e^x\vert $$</span> has a minimum.</p>
<p>This is what I did:</p>
<p><span class="math-container">$$f'(x) = (1+e^x)\text{sgn}(1+e^x)$$</span></p>
<p>But this function is never zero, because the exponential is always positive. So when I ... | Angelo | 771,461 | <p>You could also proceed in the following way.</p>
<p>By letting <span class="math-container">$\;t=x-\sqrt{1-x^2}\;,\;$</span> we get that</p>
<p><span class="math-container">$x=\dfrac12\left(t\pm\sqrt{2-t^2}\right)\;,\quad\mathrm dx=\dfrac12\left(\!\!1\mp\dfrac t{\sqrt{2-t^2}}\!\!\right)\mathrm dt\;\;.$</span></p>
<p... |
2,760,605 | <p>$||x-1|-2|=|x-3|$.
Find the value of $x$.<br>
In my attempt I got the critical values of the expression as $1$ and $3$.
But I’m not sure is we can just not consider the $2$ in the LHS.</p>
<p>My steps are Case 1: when $x>3$, $x-1-2=x-3$, $0=0$</p>
<p>Case 2: $1<x<3$, $X-1-2=-x+3 \implies 2x=6 \implies x... | adfriedman | 153,126 | <p>First, write
$$\ln\left(\frac{n}{(n!)^{1/n}}\right)
=\ln(n) - \tfrac{1}{n}\ln(n!) = \ln(n) - \tfrac1n \sum_{k=1}^n \ln(k)
= \ln(n) - \tfrac1n S_n$$</p>
<p>Consider upper and lower approximations of the integral of $\ln(x)$:</p>
<p>$$S_n = \sum_{k=1}^n \ln(k) \leq \int_1^{n+1}\ln(x)\;dx \leq \sum_{k=2}^{n+1}\ln(k) ... |
2,835,038 | <p>Assuming that sum of probabilities for all possible events that can occur should sum to 1, how does one denote this for a conditional probability? Is it $P(A|E_1) + P(A|E_2) + ... = 1$, where $E_i$ is a specific event to be conditioned on? Or is the answer something else entirely? </p>
| Shirish Kulhari | 458,802 | <p>If $E_1, E_2, \ldots$ is a collection of disjoint events whose union equals the entire sample space (exhaustive), then $P(A\cap E_1)+P(A\cap E_2)+\ldots = P(A|E_1)P(E_1)+P(A|E_2)P(E_2)+\ldots = P(A)$.</p>
|
2,871,947 | <p>I can prove that commutator is minimal subgroup such that factor group of it is abelian. I had encountered one statement as</p>
<blockquote>
<p>If <span class="math-container">$H$</span> is a subgroup containing commutator subgroup then <span class="math-container">$H$</span> is
normal.</p>
</blockquote>
<p>I.... | Angina Seng | 436,618 | <p>$G'$ is certainly normal in $G$, and $G/G'$ is Abelian. Every
subgroup of an Abelian group is normal. But $H/G'$ is a subgroup
of $G/G'$ so $H/G'$ is normal in $G/G'$. Therefore, by the third
isomorphism theorem for groups, $H$ is normal in $G$.</p>
|
780,588 | <p>I need help with this proof:</p>
<p>Let $V, W$ be K-vectorspaces.
Let $T_1, T_2$ be subspaces of V with $T_1 \subseteq T_2$.
Let $f \in hom_K(V,W)$.</p>
<p>Show the following: If $ f(T_1) = f(T_2)$ then exists a subspace $U$ of $ker(f)$ so that $ U \bigoplus T_1 = T_2$</p>
<p>I'm sorry, but I really have no clue ... | Walid Taamallah | 145,898 | <p>Not a full answer: We have that $f(T_2)=f(T_1)$ hence in particular we have the first inclusion $f(T_2)\subset f(T_1)$. For each $t_2\in T_2$ we have that $f(t_2)\in f(T_2)\implies f(t_2)\in f(T_1)$, then there exists $t_1\in T_1$ such that
$$f(t_2)=f(t_1)\implies f(t_2-t_1)=0\implies t_2-t_1\in \ker f$$
Hence t... |
2,558,646 | <p>Okay, so this is bugging me now.</p>
<p>I know this:</p>
<p>while tg angle = Y / X</p>
<p>for y > 0, angle is an element of < 0, PI ></p>
<p>for y < 0, angle is an element of < PI, 2PI ></p>
<p>for y = 0, angle is 0 or PI or 2PI.</p>
<p>Okay, but how to determine what out of those 3 values angle actua... | kimchi lover | 457,779 | <p>One way to start is to see if $f(x)/L_f(x)\sim 1$, where $L_f(x)$ is the lead term of $f(x)$.</p>
|
705,662 | <p>So my function is $f(z) = \sqrt(r)e^{i\theta/2}$.</p>
<p>I want to find its contour integral over the upper half-disk of $z = e^{i\theta}$, where $0\leq\theta\leq\pi$.</p>
<p>I know that:</p>
<p>$\int f(z) dz$ = $\int_a^b f(z(t)) * z'(t) dt$</p>
<p>But in the above problem, the function I want to find the contou... | Yiorgos S. Smyrlis | 57,021 | <p>The function $f(z)=z^{1/2}$, with $f(1)=1$, is analytic in $\mathbb C\smallsetminus \{ti: t\in (-\infty,0]\}$, and
$$
f(z)=\Big(\frac{2}{3}z^{3/2}\Big)'.
$$
So, for $\gamma: [0,\pi]\to\mathbb C$, with $\gamma(t)=\mathrm{e}^{it}$, and $g(z)=\frac{2}{3}z^{3/2}$,
$$
\int_\gamma f(z)\,dz=g\big(\gamma(\pi)\big)-g\big(\g... |
706,250 | <p>For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$.</p>
<p>I tried to set $n = 3+4k$ but it doesn't help.</p>
<p>Any hints first please?</p>
| Bill Dubuque | 242 | <p>Prime $\,P\nmid A\,\Rightarrow\,A^{\large J+K(P-1)} = A^{\large J}(\color{#c00}{\large A^{P-1}})^{\large K}\equiv A^{\large J} \color{#c00}1^{\large K}\equiv A^{\large J}\!\pmod P,\ $ by $\rm\,\color{#c00}{little\ Fermat}$.</p>
|
1,116,215 | <p>In C. Adam's Topology, it is written that $f(A) - f(B) \subset f(A - B)$ for any function, but $f(A) - f(B) = f(A - B)$ iff f is bijective. I can come the half way, i.e., $f(A) - f(B) \subset f(A - B)$; but for the $f(A - B) \subset f(A) - f(B)$, I don't know both of how to prove correctness of it for bijective and ... | Mauro ALLEGRANZA | 108,274 | <p>For the <em>bijective</em> case, consider :</p>
<blockquote>
<p><span class="math-container">$x \in f(A - B)$</span>;</p>
</blockquote>
<p>this means that there is <span class="math-container">$y_1 \in A - B$</span> such that : <span class="math-container">$x=f(y_1)$</span>, and <span class="math-container">$y_1 \in... |
3,866,673 | <p>I’m working on my linear algebra assignment and now struggling to solve this problem, say, on the complex field, find the Jordan canonical form of
<span class="math-container">$$A=\begin{bmatrix}\quad &\quad &\quad & a_1\\ \quad & \quad & a_2 & \quad \\ \quad & \dots & \quad & \qu... | 5xum | 112,884 | <p>The same way</p>
<p><span class="math-container">$$1+2+3+\cdots + 100 = 1+2+3+\cdots +99+100$$</span></p>
|
812,379 | <p>Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!</p>
| lhf | 589 | <p>By definition, $P$ is a prime ideal when $xy\in P \implies x\in P$ or $y\in P$. By induction, $x_1 x_2 \cdots x_n \in P \implies x_i \in P$, for some $i$.</p>
<p>In particular, since $0 \in P$, if $a^n=0$, then $a^n \in P$, and so $a\in P$.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.