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 |
|---|---|---|---|---|
4,307,016 | <p>Explore convergence of <span class="math-container">$\sum_{n=3}^{\infty}\frac{1}{n\ln n(\ln \ln n)^\alpha}$</span></p>
<p>Tried to use Cauchy integral test,so we need to find</p>
<p><span class="math-container">$$\int_{3}^\infty\frac{dx}{x\ln x(\ln \ln x)^\alpha}=\int_{\ln 3}^{\infty}\frac{dz}{z(\ln z)^\alpha}= \in... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Note that we have for <span class="math-container">$\alpha\ne 1$</span></p>
<p><span class="math-container">$$\int_3^\infty \frac{1}{x\log(x) \left(\log(\log(x))\right)^\alpha}\,dx=\left.\left(\frac1{(1-\alpha)\left(\log(\log(x))\right)^{\alpha-1}}\right)\right|_3^\infty$$</span></p>
<p... |
1,553,354 | <p>Help me to find an example of a sequence of differentiable functions defined on $[0,1]$ that converge uniformly to a function $f$ on $[0,1]$ such that there exists $x \in (0,1)$ such that $f$ is not differentiable at $x$.</p>
| Christian Blatter | 1,303 | <p>Consider the $C^1$-functions
$$f_n(x):=\sqrt{{1\over n^2}+x^2}-{1\over n}\qquad(n\geq1)\ .$$
One has $f_n(0)=0$ for all $n$, and
$$f_n(x)={x^2\over \sqrt{{1\over n^2}+x^2}+{1\over n}}\to |x|\qquad(n\to\infty)$$
for all $x\ne0$. It follows that $\lim f_n(x)=|x|$ for all $x\in{\mathbb R}$, which is not differentiable ... |
1,162,697 | <p>If $f:\mathbb{R} \to \mathbb{R}$ is differentiable with at least two roots, I wish to show that Newton's method will not converge for some $x_0$. </p>
<p>I know that $f'(x)$ has a zero, say at $z$. It seems we should choose $x_0$ close to $z$ to ensure that the Newton iterates wander away. But it's hard to say ... | Understand | 214,109 | <p>Have a look at this example:
$$x^3 - 5x = 0$$
$$x_0=1$$</p>
|
1,162,697 | <p>If $f:\mathbb{R} \to \mathbb{R}$ is differentiable with at least two roots, I wish to show that Newton's method will not converge for some $x_0$. </p>
<p>I know that $f'(x)$ has a zero, say at $z$. It seems we should choose $x_0$ close to $z$ to ensure that the Newton iterates wander away. But it's hard to say ... | Robert Israel | 8,508 | <p>If $f'(x_0) = 0$, there is no $x_1$.<br>
In the case of a quadratic $f$ with two real roots, that is the only initial point where Newton goes wrong: for all other $x_0$, it converges to one of the roots.</p>
<p>EDIT: Somewhat more generally, if $f$ is a convex differentiable function with at least one root, you hav... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | GEdgar | 442 | <p>Here's what I got from the equation using Maple... </p>
<p><img src="https://i.stack.imgur.com/6pXe9.jpg" alt="enter image description here"></p>
|
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | Shivam Patel | 95,509 | <p>Sorry but this is not the answer but too long for a comment:
Probably the easiest verification is to type the equation on Google you'l be surprised :
The easiest way is to Google :2 sqrt(-abs(abs(x)-1)<em>abs(3-abs(x))/((abs(x)-1)</em>(3-abs(x))))(1+abs(abs(x)-3)/(abs(x)-3))sqrt(1-(x/7)^2)+(5+0.97(abs(x-.5)+abs(x+.5... |
939,509 | <p>Is there a proper name for a shape defined by the volume between two concentric spheres? My understanding is that, formally, a "sphere" is strictly a 2D surface and there's a formal term for volume contained by that surface -- which I forget.</p>
<p>Is there a term that describes the volume between two concentric s... | Jonas Meyer | 1,424 | <p>It's also called an <a href="http://mathworld.wolfram.com/AnnulusTheorem.html" rel="nofollow">annulus</a>. ${}$</p>
|
2,461,615 | <p>I am still at college. I need to solve this problem.</p>
<p>The total amount to receive in 1 year is 17500 CAD.
And the university pays its students each 2 weeks (26 payments per year). </p>
<p>How much does a student have to receive for 4 months?
I have calculated this in 2 ways (both seem ok) but results are di... | Aizzaac | 488,697 | <p>Okey. This is my solution:</p>
<pre><code>1 year = 365 days or 366 days
two-week period = 14 days
365 / two-week period = 26 payments per year
September to December = 122 days
122 / two-week period = 8.71 payments
(175000 / 26) x 8.71 = 5863 CAD
</code></pre>
|
3,393,244 | <p>My homework is to transform this formula </p>
<p><span class="math-container">$$(A \wedge \neg B) \wedge (A \vee \neg C)$$</span> into this equivalent form: <span class="math-container">$A \wedge \neg B$</span>. Do you have any ideas?</p>
| Bram28 | 256,001 | <p>The 'correct' transformation depends on what rules you have .... </p>
<p>Here is a transformation that uses pretty elementary equivalence principles:</p>
<p><span class="math-container">$$(A \wedge \neg B) \wedge (A \vee \neg C)$$</span></p>
<p><span class="math-container">$$\overset{Commutation}{=}$$</span></p>... |
2,216,601 | <p>Alright so I have this Transformation that I know isn't one to one transformation, but I'm not sure why. </p>
<p>A Transformation is defined as $f(x,y)=(x+y, 2x+2y)$.</p>
<p>Now my knowledge is that you need to fulfill the 2 conditions: Additivity and the scalar multiplication one. I tried both of them and somehow... | Jimmy R. | 128,037 | <p><strong>Hint:</strong> What is $f^{-1}(0,0)$? For example $f(0,0)=(0,0)$, so $(0,0)\in f^{-1}(0,0)$. Can you find (m)any other pair(s) $(x,y)$ such that $f(x,y)=(0,0)$? </p>
|
777,535 | <p>I need to find the full Taylor expansion of $$f(x)=\frac{1+x}{1-2x-x^2}$$</p>
<p>Any help would be appreciated. I'd prefer hints/advice before a full answer is given. I have tried to do partial fractions\reductions. I separated the two in hopes of finding a known geometric sum but I could not.</p>
<p>Edit: I guess... | Madavan Viswanathan | 547,205 | <p>Consider the following, doing the <span class="math-container">$2$</span>-D case, which can be generalized to <span class="math-container">$n$</span>-D.</p>
<p>Vector <span class="math-container">$A$</span> with coordinates <span class="math-container">$(x_A,y_A)$</span></p>
<p>Vector <span class="math-container">$B... |
4,394,983 | <p>I am tasked with proving that Th((<span class="math-container">$\mathbb{Z}, <))$</span> has continuum many models.
For this we are given the following construction.</p>
<blockquote>
<p>Let <span class="math-container">$\alpha \in \mathcal{C} = \{0,1\}^{\mathbb{N}}$</span>. We define for each <span class="math-con... | Primo Petri | 137,248 | <p>This is not exactly the answer that the OP is looking for, but it may be interesting.</p>
<p>First an answer that assumes the continuum hypothesis.</p>
<p>For every <span class="math-container">$\alpha<\omega_1\smallsetminus\{0\}$</span> there is a model <span class="math-container">$\alpha\times \mathbb Z$</span... |
3,424,656 | <p>Assume <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous at <span class="math-container">$x=a$</span>. Prove <span class="math-container">$h=\max\{f,g\}$</span> is continuous at <span class="math-container">$x=a$</span>.</p>
<p>My solution:</p>
<p>When <span class="... | José Carlos Santos | 446,262 | <p>No, not at all. By the same argument, if <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are differentiable, then so is <span class="math-container">$\max\{f,g\}$</span>. However, <span class="math-container">$\max\{x,-x\}=\lvert x\rvert$</span>.</p>
|
4,569,910 | <p><span class="math-container">$ABC$</span> is a right-angled triangle (<span class="math-container">$\measuredangle ACB=90^\circ$</span>). Point <span class="math-container">$O$</span> is inside the triangle such that <span class="math-container">$S_{ABO}=S_{BOC}=S_{AOC}$</span>. If <span class="math-container">$AO^2... | Sarvesh Ravichandran Iyer | 316,409 | <p>This is just a more rigorous write-up of things, because there is a more general phenomena here that is in play : an exchange argument.</p>
<p>Summary :</p>
<ul>
<li><p>The "success probability" calculation</p>
</li>
<li><p>The idea of an exchange argument.</p>
</li>
<li><p>Using the exchange argument to s... |
18,511 | <p>I have a notebook written in Mathematica 8 in which I imported Tiff images and everything worked fine. Since I installed Mathematica 9, I get the error:</p>
<pre><code>In[14]:= Files[[1]][[1]]
Import[Files[[1]][[1]],"TIFF"]
Out[14]= Growth_1_130124_1353/Growth_1_130124_1353_T0001.tif
During evaluation of In[14]:= I... | Sjoerd C. de Vries | 57 | <p>Photoshop complains that the ICC color profile of this picture is invalid and is ignoring it. So, there might be a problem with the picture itself. It is a 1-channel grayscale image but it reports a 3-channel RGB color space.</p>
<p>Mathematica 9 has the new <code>ColorProfileData</code> object which represents thi... |
333,467 | <p>I was reading in my analysis textbook that the map $ f: {\mathbf{GL}_{n}}(\mathbb{R}) \to {\mathbf{GL}_{n}}(\mathbb{R}) $ defined by $ f(A) := A^{-1} $ is a continuous map. I also saw that $ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $. My question is:</p>
<blockquote>
<p>What is t... | user1551 | 1,551 | <p>As pointed out by the others, you cannot extend $f$ to a <em>continuous</em> function $g:M_n(\mathbb{R})\to M_n(\mathbb{R})$, because there exists a convergent sequence of invertible matrices $X_n$ such that $f(X_n)=X_n^{-1}$ diverges. There does exist, however, a <em>bijective</em> function $g:M_n(\mathbb{R})\to M_... |
1,677,359 | <p>$\sum_{i=0}^n 2^i = 2^{n+1} - 1$</p>
<p>I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks</p>
| Slade | 33,433 | <p>Since you asked about Pascal's triangle:</p>
<p>Imagine filling in rows $0$ through $n$ of Pascal's triangle. Now change the first position of row $0$ from $1$ to $1+1$.</p>
<p>Distribute the two ones to the following row, which should now read $1+1, 1+1$. Distribute again to get $1+1,2+2,1+1$. And so on.</p>
... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | YukiJ | 7,608 | <p>The <a href="https://en.wikipedia.org/wiki/Minimax_theorem" rel="noreferrer">minimax theorem</a> states the following:</p>
<blockquote>
<p>Let $X\subset \mathbb{R}^{n}$ and $Y\subset \mathbb {R} ^{m}$ be
compact convex sets. If $ f:X\times Y\rightarrow \mathbb {R} $ is a
continuous function that is convex-c... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Torsten Schoeneberg | 8,931 | <p>Last year I heard of </p>
<p>$$\text{Lo De Hi Mi Hi De Lo}$$</p>
<p>$$\text{(sing: "Low Dee High my High Dee Low!")}$$</p>
<p>as a mnemonic for the numerator in the quotient rule:</p>
<p>$$\left(\frac{f}{g}\right)' = \frac{g\cdot Df - f \cdot Dg}{g^2}$$</p>
<p>(of course Lo(w) = denominator, De = derivative, Hi... |
2,064,095 | <p>Can someone please help me understand this problem
Does the limit exist in the part (a) and part (b)</p>
<p>A) $$\lim_{(x,y) \to (0,0)} x \sin (\frac{1}{y})$$</p>
<p>B) $$\lim_{(x,y) \to (0,0)} \left( x \sin (\frac{1}{y})+y \sin (\frac{1}{x}) \right)$$</p>
| Henricus V. | 239,207 | <p>Problem 1 only holds for finite $A$. Since $x \not\in A \implies \chi_A(x) = 0$, the sum reduces to $\sum_{x \in X} \chi_A(x) = \sum_{x \in A} \chi_A(x) = \sum_{x \in A} 1 = |A|$.</p>
<p>Problem 2 is essentially the <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferr... |
2,064,095 | <p>Can someone please help me understand this problem
Does the limit exist in the part (a) and part (b)</p>
<p>A) $$\lim_{(x,y) \to (0,0)} x \sin (\frac{1}{y})$$</p>
<p>B) $$\lim_{(x,y) \to (0,0)} \left( x \sin (\frac{1}{y})+y \sin (\frac{1}{x}) \right)$$</p>
| Brian M. Scott | 12,042 | <p>For the first question you have simply</p>
<p>$$|A|=\sum_{x\in A}1=\sum_{x\in A}1+\sum_{x\in X\setminus A}0=\sum_{x\in X}\chi_A(x)\;,$$</p>
<p>assuming, of course, that $A$ is a finite set.</p>
<p>HINT: For the second question just compare the two sides for each $x\in X$. Note that each $x\in X$ is in exactly one... |
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | rm -rf | 5 | <p>The simplest way would be to replace zeros with <code>Null</code>, map <code>Sort</code> onto it and then replace <code>Null</code> with zeros. This works because the default sorting function <code>OrderedQ</code> will place <code>Null</code> at the end, as per your needs.</p>
<pre><code>mat = {{0, 0, 3.411, 0, 1.... |
1,656,136 | <p>I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)</p>
| spec | 318,677 | <p>Consider the ring $R = C([0,1], \mathbf{R})$ of continuous functions $f \colon [0,1] \to \mathbf{R}$.</p>
<p>For $0 \lt t \lt 1$, let $I_t = \{ f \in R \mid f(x) = 0 \text{ for all } 0 \leq x \leq t \}$ be the ideal of functions vanishing on all of $[0,t]$. Then $I_s \supsetneq I_t$ whenever $s \lt t$.</p>
|
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We have</p>
<p>$$\sqrt a + \sqrt b=c \stackrel{both \, terms\, \ge 0}\iff (\sqrt a + \sqrt b)^2=a+2\sqrt{ab}+b=c^2 $$</p>
<p>and</p>
<p>$$a+2\sqrt{ab}+b=c^2 \color{red}{\implies} (2\sqrt{ab})^2=(c^2-a-b)^2$$</p>
<p>for the latter implication we need to check at the end for possible ... |
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | mfl | 148,513 | <p><strong>First step</strong></p>
<p>$$3\sqrt{x-1}+\sqrt{3x+1}=2\implies (3\sqrt{x-1}+\sqrt{3x+1})^2=2^2.$$</p>
<p><strong>Second step</strong></p>
<p>After rearranging you'll get</p>
<p>$$6\sqrt{x-1}\sqrt{3x+1}=ax+b.$$ Take squares one more time.</p>
<p><strong>Final step</strong></p>
<p>Solve the quadratic equ... |
206,305 | <p>Prove: $s_n \to s \implies \sqrt{s_n} \to \sqrt{s}$ by the definition of the limit. $s \geq 0$ and $s_n$ is a sequence of non-negative real numbers.</p>
<p>This is my preliminary computation:</p>
<p>$|\sqrt{s_n} - \sqrt{s}| < \epsilon$</p>
<p>multiply by the conjugate:</p>
<p>$|\dfrac{s_n - s}{\sqrt{s_n}+\sqr... | Pragabhava | 19,532 | <p>If both $s$ and $s_n$ are non-negative</p>
<p>$$
|\sqrt{s}-\sqrt{s_n}|^2 \le |\sqrt{s}-\sqrt{s_n}||\sqrt{s} + \sqrt{s_n}|.
$$</p>
<p><strong>Step by Step :)</strong></p>
<p>Since both $s$ and $s_n$ are non-negative</p>
<p>$$
|\sqrt{s}-\sqrt{s_n}| \le |\sqrt{s} + \sqrt{s_n}|
$$</p>
<p>this is clear because the r... |
99,506 | <p>I am trying to show that how the binary expansion of a given positive integer is unique.</p>
<p>According to this link, <a href="http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf" rel="nofollow">http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf</a>, All I see is that I can recopy theorem 3-1's pro... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Put $\ b_i = 2\ $ in this sketched proof of the uniqueness of <a href="http://en.wikipedia.org/wiki/Mixed_radix" rel="nofollow">mixed-radix representation</a></p>
<p>$$\begin{eqnarray} n &=& d_0 +\ d_1\, b_0 +\ d_2\, b_1\,b_0 +\ d_3\, b_2\,b_1\,b_0 +\ \cdots, \quad 0 \le d_i < ... |
981,541 | <p>Say I have the number <code>0.73992</code> and I'm rounding to 3 decimal places. My instinct would be to write <code>0.740 (3dp)</code>. But surely that implies that it is <em>exactly</em> <code>0.740</code>. The only other alternatives are to write <code>0.7399 (4dp)</code> or <code>0.74</code>, neither of which ar... | ACupofJoe | 185,680 | <p>.0740 would be the answer. Since you're adding the 0 at the end it implies that you have an accuracy to the third decimal place.</p>
|
981,541 | <p>Say I have the number <code>0.73992</code> and I'm rounding to 3 decimal places. My instinct would be to write <code>0.740 (3dp)</code>. But surely that implies that it is <em>exactly</em> <code>0.740</code>. The only other alternatives are to write <code>0.7399 (4dp)</code> or <code>0.74</code>, neither of which ar... | Barbosa | 185,459 | <p>If it was 0.11488 you would round to 0.115, and it does not implies that it is exactly 0.115. I think you just confused because the rounded number ended on 0. So, in your example, you should write 0.740.</p>
|
547,050 | <p>Which trigonometric formulas are used for these problems?
<img src="https://i.stack.imgur.com/TVBCx.png" alt="enter image description here"></p>
| Empy2 | 81,790 | <p>These come from the addition formulas $\cos(a+b)+\cos(a-b)=2\cos a\cos b$ and $\sin(a+b)+\sin(a-b)=2\sin(a)\cos(b)$</p>
|
831,472 | <p>I am learning about Karnaugh maps to simplify boolean algebra expressions. I have this:</p>
<p>$$\begin{bmatrix}
& bc & b'c & bc' & b'c' \\
a & 0 & 1 & 1 & 0\\
a' & 1 & 1 & 0 & 1
\end{bmatrix}$$</p>
<p>There are no groups of four, so I am now looking for groups of tw... | skyking | 265,767 | <p>As pointed out you need to order the rows and columns properly so that adjacent cells differ only in one variable. If your labeling is correct you need to swap the rightmost two columns for this (then of course it's customary to order the columns 00, 01, 11 and 10, but that's not necessary for the working of the dia... |
203,111 | <p>Assume $(A_{i})_{i\in\Bbb N}$
to be an infinite sequence of sets of natural numbers, satisfying</p>
<p>$$A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq A_{3}\cdots\subseteq\Bbb N\tag{*}$$</p>
<p>For each property $p_{i}$
shown below, state whether </p>
<p>• the hypothesis (*)
is sufficient to conclude that $p... | Berci | 41,488 | <p>($*$)$\Rightarrow p_1,p_6$. </p>
<p>The rest, $p_2, p_3, p_4, p_5$ are not in general true, even if we assume (*). Try to find counterexamples.</p>
|
498,785 | <p>I'm trying to solve this problem, but I'm not even sure how to formulate it in a coherent mathematical manner, or even what branch of mathematics this might fall in to.</p>
<p>Basically I have a set of weights, where each weight individually must remain in the range $[0,1]$. I want to change the mean of the weight... | Sneftel | 10,735 | <p>First, put aside all the 0s and 1s, which will stay the same. (If you only have 0s and 1s you'll need to use a different strategy, and you won't be able to do #3.) Put the remaining weights through the logit function. Then find a constant which you can add to all the logit-scale weights such that, after putting them... |
63,633 | <p>(This question came up in a conversation with my professor last week.)</p>
<p>Let $\langle G,\cdot \rangle$ be a group. Let $x$ be an element of $G$.
<br>
Is there always an isomorphism $f : G \to G$ such that $f(x) = x^{-1}$ ?
<br>
What if $G$ is finite?</p>
| GH from MO | 11,919 | <p>The Mathieu group $M_{11}$ does not have this property. A quote from Example 2.16 in <a href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3895v1.pdf">this paper</a>: "Hence there is no automorphism of $M_{11}$ that maps $x$ to $x^{−1}$."</p>
<p>Background how I found this quote as I am no group theorist: I used G... |
63,633 | <p>(This question came up in a conversation with my professor last week.)</p>
<p>Let $\langle G,\cdot \rangle$ be a group. Let $x$ be an element of $G$.
<br>
Is there always an isomorphism $f : G \to G$ such that $f(x) = x^{-1}$ ?
<br>
What if $G$ is finite?</p>
| Tim Dokchitser | 3,132 | <p>No, such an isomorphism does not always exist, and the smallest counterexample is $G=C_5\rtimes C_4$ with $C_4$ acting faithfully. It is not hard to see that the only automorphisms of $G$ are inner, and that they cannot map an element of order 4 to its inverse.</p>
|
2,512,736 | <p>I do not understand how this result is a special case of theorem 9.1, could anyone explain this for me please?</p>
<p><a href="https://i.stack.imgur.com/hsgYr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hsgYr.png" alt="enter image description here"></a></p>
<p>This is theorem 9.1:</p>
<p><a... | greg | 357,854 | <p>Let a lowercase letter stand for the corresponding vectorized matrix, e.g.
$$f={\rm vec}(F), \,\,\, x={\rm vec}(X)$$
Write the function and its differential
$$\eqalign{
F &= A^TX + XA - XPX + Q \cr
dF &= A^T\,dX + dX\,A - dX\,PX -XP\,dX \cr
}$$
and vectorize
$$\eqalign{
df &= (I\otimes A^T+A^T\otimes I-... |
94,440 | <p>In Sean Carroll's <em>Spacetime and Geometry</em>, a formula is given as
$${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$$</p>
<p>where $K^\mu$ is a Killing vector satisfying Killing's equation ${\nabla _\mu }{K_\nu } +{\nabla _\nu }{K_\mu }=0$ and the convention of Riemann curvatu... | K.defaoite | 553,081 | <p>A solution that is simpler still, requiring no differentiation or fancy identities other than one of the Bianchi identities, is as follows.</p>
<hr />
<p>We know that the Riemann tensor can measure how much the covariant derivatives commute with each other, e.g
<span class="math-container">\begin{equation}
[ \nabla ... |
1,677,868 | <p>The sequence is:</p>
<p>$$a_n = \frac {2^{2n} \cdot1\cdot3\cdot5\cdot...\cdot(2n+1)} {(2n!)\cdot2\cdot4\cdot6\cdot...\cdot(2n)} $$</p>
| Claude Leibovici | 82,404 | <p>In the same spirit as Brian M. Scott's answer, using $$2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!$$ and $${1\cdot 3\cdot 5\cdots (2n+1)}=\frac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdots (2n+1)}{2\cdot4\cdot6\cdots(2n)}=\frac{(2n+1)!}{2^n n! }$$ All of this makes $$a_n=\frac{2^{2n}\frac{(2n+1)!}{2^n n!}}{2^n n! (2n)!}=\frac{... |
365,483 | <p>Let <span class="math-container">$f\colon X\to \mathbb{A}^n_{\mathbb{C}}$</span> be a morphism of <span class="math-container">$\mathbb{C}$</span>-schemes. Suppose <span class="math-container">$f$</span> is (a) separated, (b) flat, (c) locally of finite type, (d) all fibers are quasi-compact, is <span class="math-co... | R. van Dobben de Bruyn | 82,179 | <p>Here is a counterexample:</p>
<p><strong>Example.</strong> We will define <span class="math-container">$X$</span> as a union of affine varieties
<span class="math-container">$$U_0 \subseteq U_1 \subseteq \ldots$$</span>
as follows: start with <span class="math-container">$U_0 = \mathbf A^1 \times (\mathbf A^1 \setmi... |
330,991 | <p>Many things in math can be formulated quite differently; see the list of statements equivalent to RH <a href="https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis">here</a>, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, et... | Mohammad Al-Turkistany | 8,784 | <p>The P vs NP problem can be formulated in terms of <em>incomplete</em> sets in NP. Ladner theorem can be stated as:</p>
<p><span class="math-container">$P \ne NP$</span> if and only if there is an incomplete set in NP.</p>
<p>Incomplete set is a set that is not complete for <span class="math-container">$NP$</span> ... |
338,535 | <p>Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$.
Given that $f(1)= 2013$, find the value of $f(2013)$.</p>
| Christian Blatter | 1,303 | <p>Introduce the auxiliary function
$$g(n):=n^2 f(n)\qquad(n\geq1)\ .$$
Then
$$n g(n)= g(1)+g(2)+\ldots+g(n)\qquad(n\geq1)$$
and therefore
$$(n+1)g(n+1)-n g(n)=g(n+1)\ ,$$
or $g(n+1)=g(n)$ for all $n\geq1$. It follows that
$$2013^2 f(2013)= g(2013)= g(1)=1^2 f(1)\ ,$$
whence $f(2013)={1\over2013}$.</p>
|
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | mathreadler | 213,607 | <p>The purpose of rigor is not so much to make sure something is true. It is to make sure we know what we are actually assuming. If one forces specificity of what is assumed then also new ways to define thing may become clearer. </p>
<p>The parallell axiom of euclidean geometry is a good example. By forcing ourselves ... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | Philip Roe | 430,997 | <p>@TheGreatDuck This is a fascinating thread. Let me comment on your aeronautical contribution from the viewpoint of an aeronautical engineer.</p>
<p>There are times when rigor is important and times when it isnt. For the first situation, consider the design of software to undertake air traffic control. Much attentio... |
3,489,212 | <p>Playing around I found a series which looks to converge to the square root function.</p>
<p><span class="math-container">$$\sqrt{p^2+q}\overset{?}{=}p\left(1-\sum_{n=1}^{+\infty}\left(-\frac q{2p^2}\right)^n\right)$$</span></p>
<p>Is it correct?</p>
| kimchi lover | 457,779 | <p>No, it is not correct. Your right-hand expression is a geometric series for <span class="math-container">$$p\left(1-\frac{-\frac q{2p^2}}{1+\frac q{2p^2}}\right),$$</span>
which is a rational expression in <span class="math-container">$p$</span> and <span class="math-container">$q$</span>.</p>
<p>The correct answe... |
364,278 | <p>Let <span class="math-container">$X$</span> be a variety over a number field <span class="math-container">$K$</span>. Then it is known that for any topological covering <span class="math-container">$X' \to X(\mathbb{C})$</span>, the topological space <span class="math-container">$X'$</span> can be given the structur... | Will Chen | 15,242 | <p>Here's a simple argument assuming <span class="math-container">$X$</span> admits a <span class="math-container">$K$</span>-rational point, and that <span class="math-container">$X$</span> has a finitely generated geometric fundamental group. In fact the "further" covering <span class="math-container">$X''$... |
3,453,408 | <p>I'm reading through some lecture notes and see this in the context of solving ODEs:
<span class="math-container">$$\int\frac{dy}{y}=\int\frac{dx}{x} \rightarrow \ln{|y|}=\ln{|x|}+\ln{|C|}$$</span> why is the constant of integration natural logged here?</p>
| Fimpellizzeri | 173,410 | <p>No real reason, from this simple equation, that I can see. It could be any <span class="math-container">$C$</span>. Perhaps the author intended to take the exponential of both sides in the following step and remind you that in this case the constant term must be non-negative.</p>
|
155,547 | <p>Given $X_1, \ldots, X_n$ from $\mathcal{N} (\mu, \sigma^2)$.</p>
<p>I have to compute the probability:
$$P\left(|\bar{X} - \mu| > S\right)$$
where $\bar{X}$ is the sample mean and $S^2$ is the sample variance.</p>
<p>I tried to expand:
$$P\left(\bar{X}^2 + \mu^2 - \bar{X}\mu > \frac{1}{n}\sum {X_i}^2 + \frac... | Michael Hardy | 11,667 | <p>$$
\frac{\bar X - \mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1)
$$
$$
\frac{\bar X - \mu}{S/\sqrt{n}} \sim T_{n-1}
$$
where $T_k$ is Student's t-distribution with $k$ degrees of freedom.</p>
<p>So
$$
\Pr\left(\left|\frac{\bar X - \mu}{S}\right| > 1\right) = \Pr\left(\left|\frac{\bar X - \mu}{S/\sqrt{n}}\right| >... |
4,021,994 | <p>I was taught in high school algebra to translate word problems into algebraic expressions. So when I encountered <a href="https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_3" rel="nofollow noreferrer">this</a> problem I tried to reason out an algebra formula for it</p>
<blockquote>
<p>For ... | Laowl Lomao | 851,165 | <p>You can use linear algebra, by defining something as x, say the amount paid by Ben. Then Dave's will be 0.75x. Then we can get an easily solvable equation.</p>
|
4,021,994 | <p>I was taught in high school algebra to translate word problems into algebraic expressions. So when I encountered <a href="https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_3" rel="nofollow noreferrer">this</a> problem I tried to reason out an algebra formula for it</p>
<blockquote>
<p>For ... | poetasis | 546,655 | <p>If the difference is <span class="math-container">$\$12.50$</span> then it must be the product of dollars times <span class="math-container">$\$0.25$</span> so we divide:
<span class="math-container">$\quad\dfrac{12.50}{0.25}=50\quad$</span> for Ben and another <span class="math-container">$\quad 37.50 \quad$</span>... |
403,631 | <p>$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
<br/>
Hint $u_{2n}$ = $u_{n}^2$</p>
<p>I have totally no idea how to prove this, this looks obvious but i found out proof is really hard...
I am doing a real analysis course and there's a lot of proving and I stuck there.
Any advices? Pra... | robjohn | 13,854 | <p>Since $0\le|a|\lt1$, we have $0\le|a|^{n+1}\le|a|^n$. Since $|a|^n$ is a non-increasing sequence, bounded below, $A=\lim\limits_{n\to\infty}|a|^n$ exists. Then,
$$
\begin{align}
|a|A
&=|a|\lim_{n\to\infty}|a|^n\\
&=\lim_{n\to\infty}|a|^n\\
&=A
\end{align}
$$
Thus, $(|a|-1)A=0\implies A=0$. Therefore,
$$
... |
319,262 | <p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the circle that have a sum greater than or equal to 17? </p>
<p>This was from a textbook called "Discrete math and its application", however it does not provide solution for this question... | Steve Kass | 60,500 | <p>Remove the number 1 and unwrap the circle of numbers into a row $a, b, c, d, e, f, g, h, i$, where $\{a, b, c, d, e, f, g, h, i\}=\{2,3,4,5,6,7,8,9,10\}$. Then $(a+b+c)+(d+e+f)+(g+h+i)=\sum_{j=2}^{10}j = 54$, therefore at least one of $(a+b+c), (d+e+f),$ or $(g+h+i)$ must be $\ge {54\over3}=18$. </p>
|
599,126 | <p>Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$.</p>
<ul>
<li>$f$ has to be uniformly continuous.</li>
<li>there exists a $x\in \mathbb{R}$ such that $f(x)=x$.</li>
<li>$f$ can not be increasing.</li>
<li>$\lim_{x... | Eric Auld | 76,333 | <p>$\tan^{-1}x$ is increasing. $\sin (x^3)$ has no limit at infinity.</p>
|
4,244,187 | <blockquote>
<p>Find the equation of the tangent line to <span class="math-container">$\sin^{-1}(x) + \sin^{-1}(y) = \frac{\pi}{6}$</span> at the point <span class="math-container">$(0,\frac{1}{2})$</span></p>
</blockquote>
<p>This is in the context of learning implicit differentiation.</p>
<p>First, I apply <span clas... | Stefan | 965,450 | <p>I have found now a counterexample that shows that the considered inequality does generally not hold. Specifically, if B and C are generated as <span class="math-container">$B = Q_B Q_B^*$</span> and <span class="math-container">$C = Q_C Q_C^*$</span>, where <span class="math-container">$Q_B$</span> and <span class="... |
1,924,033 | <blockquote>
<p><strong>Question.</strong> Let $\mathfrak{g}$ be a real semisimple Lie algebra admitting an invariant inner-product. Is every connected Lie group with Lie algebra $\mathfrak{g}$ compact?</p>
</blockquote>
<p>I know that the converse is true: If $G$ is a compact connected Lie group, then the Haar meas... | Ben | 12,885 | <p>Here comes a diagram in $\mathrm(Top)$ whose colimit is the desired glueing. Given a glueing datum as in the question text, the diagram will consist of </p>
<ul>
<li>all $U_i$ and all $U_{ij}$ as objects
and </li>
<li>the inclusion maps $U_{ij}\to U_i$ as well as all $\varphi_{ij}$ as morphisms.</li>
</ul>
<p>This... |
1,512,171 | <p>I want to show that there exists a diffeomorphic $\phi$ such that the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
TS^1 @>{\phi}>> S^1\times\mathbb{R}\\
@V{\pi}VV @V{\pi_1}VV \\
S^1 @>{id_{S^1}}>> S^1
\end{CD}$$
where $\pi$ is the associated projection of $TS^1$, and $\pi_1(x,y)=x$ is ... | Ross Millikan | 1,827 | <p>It depends on your definition of integral. The Riemann integral, the first one taught in calculus classes, does not have a value because the lower sum is always zero and the upper sum is always one. The <a href="https://en.wikipedia.org/wiki/Lebesgue_integration" rel="nofollow">Lebesgue integral</a> of this functi... |
1,341,440 | <p>I came across a claim in a paper on branching processes which says that the following is an <em>immediate consequence</em> of the B-C lemmas:</p>
<blockquote>
<p>Let $X, X_1, X_2, \ldots$ be nonnegative iid random variables. Then $\limsup_{n \to \infty} X_n/n = 0$ if $EX<\infty$, and $\limsup_{n \to \infty} X_... | Tobias Kildetoft | 2,538 | <p>Since $Z(G)$ is not trivial, it has order at least $2$. But the quotient $G/Z(G)$ is not cyclic unless $Z(G) = G$ (the quotient by the center is never non-trivial cyclic), so it must have exponent dividing $2$, which precisely means that for any $x\in G$ we have $x^2\in Z(G)$.</p>
|
276,329 | <p>I have a problem, from Gelfand's "Algebra" textbook, that I've been unable to solve, here it is:</p>
<p><strong>Problem 268.</strong> </p>
<p>What is the possible number of solutions of the equation $$ax^6+bx^3+c=0\;?$$</p>
<p>Thanks in advance.</p>
| amWhy | 9,003 | <p><strong>Hint:</strong> $\quad$Let $y = x^3$:</p>
<p>$$ax^6 + bx^3 + c = 0 \quad \iff \quad ay^2 + by + c = 0\tag{1}$$</p>
<p>Solve for $y$ ... there will be either two real solutions, one real solution, or no real solutions when solving for $y$ (why?, when?). (Examine the <a href="http://en.wikipedia.org/wiki/Disc... |
2,574,117 | <p>For a matrix $A$, define the operator $\ell_p$-norm of $A$ to be
$$
\|A\|_p = \sup_{x \neq 0} \frac{\|Ax\|_p}{\|x\|_p}.
$$
Here $\|x\|_p$ denotes the $\ell_p$ norm of the vector $x$.</p>
<p>For $1 \le p \le q \le 2$ and $x \in \mathbb{R}^n$, we know that $\|x\|_q \le \|x\|_p \le n^{1 / p - 1 / q} \|x\|_q$. </p>
<p... | Martin Argerami | 22,857 | <p>Your estimate is not naive in general. With $p=1$, $q=2$, take
$$
A=\begin{bmatrix}
1&0&\cdots&0\\
1&0&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
1&0&\cdots&0\\
\end{bmatrix},
$$
It is well-known that
$$\|A\|_1=\max\{\|A_j\|_1:\ j\},\ \ \ \|A\|_2=\|A^*A\|_2^{1/2}=\... |
3,978,303 | <p><strong>Background</strong></p>
<p>The following Euler product for the Riemann zeta function is well known.</p>
<p><span class="math-container">$$ \sum_n \frac{1}{n^s} = \prod_p (1-\frac{1}{p^s})^{-1} $$</span></p>
<p>Here <span class="math-container">$n$</span> ranges over all integers, <span class="math-container"... | Paul Sinclair | 258,282 | <p>By definition, where <span class="math-container">$p_i$</span> is the <span class="math-container">$i^{th}$</span> prime,
<span class="math-container">$$\prod_{p_i} \left(1-\frac{1}{p_i^s}\right)^{-1}:=\prod_{i=1}^\infty\left(1-\frac{1}{p_i^s}\right)^{-1}:= \lim_N \prod_{i=1}^N\left(1-\frac{1}{p_i^s}\right)^{-1}$$</... |
2,098,395 | <p>Evaluate the following;</p>
<p>$$\sum_{r=0}^{50} (r+1) ^{1000-r}C_{50-r}$$</p>
<p>Using $^{n}C_{r}=^{n}C_{n-r}$ we get $\sum_{r=0}^{50} (r+1) ^{1000-r}C_{950}$</p>
<p>but I am not getting how to solve $\sum_{r=0}^{50} r \cdot \hspace{0.5 mm} ^{1000-r}C_{950}$</p>
| lab bhattacharjee | 33,337 | <p>Set $50-r=u$
$$\sum_{u=0}^{50}(51-u)\binom{950+u}{950}=\sum_{u=0}^{50}\{1002-(951+u)\}\binom{950+u}{950}$$</p>
<p>$$=1002\sum_{u=0}^{50}\binom{950+u}{950}-951\sum_{u=0}^{50}\binom{951+u}{951}$$</p>
<p>Now $\displaystyle\sum_{u=0}^{50}\binom{950+u}{950}$ is the coefficient of $x^{950}$ in $$\displaystyle\sum_{u=0}... |
2,098,395 | <p>Evaluate the following;</p>
<p>$$\sum_{r=0}^{50} (r+1) ^{1000-r}C_{50-r}$$</p>
<p>Using $^{n}C_{r}=^{n}C_{n-r}$ we get $\sum_{r=0}^{50} (r+1) ^{1000-r}C_{950}$</p>
<p>but I am not getting how to solve $\sum_{r=0}^{50} r \cdot \hspace{0.5 mm} ^{1000-r}C_{950}$</p>
| Mike Earnest | 177,399 | <p>This is a hockey stick made of hockey sticks. Expand each term <span class="math-container">$(r+1)\binom{1000-r}{950}$</span> into a column of <span class="math-container">$r+1$</span> copies of <span class="math-container">$\binom{1000-r}{950}$</span>, then add up the rows using the hockey stick identity, then add ... |
168,053 | <p>If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being concave and then becomes convex from a point on, that g does not have to be convex, but can someone show me an example... | Brian M. Scott | 12,042 | <p>Try $$f(x)=\frac{\pi}2-\tan^{-1}x\;.$$</p>
|
139,021 | <p>Can you, please, recommend a good text about algebraic operads?</p>
<p>I know the main one, namely, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Loday, Vallette "Algebraic operads"</a>. But it is very big and there is no way you can read it fast. Also there are no... | David White | 11,540 | <p><a href="http://math.univ-lille1.fr/~fresse/OperadModuleFunctors-Updated.pdf" rel="nofollow">Benoit Fresse's book <em>Modules over Operads and Functors</em></a> is masterful.</p>
<p>Additionally, here are a couple of very good survey articles and notes from conferences:</p>
<p><a href="http://www.ams.org/notices/2... |
139,021 | <p>Can you, please, recommend a good text about algebraic operads?</p>
<p>I know the main one, namely, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Loday, Vallette "Algebraic operads"</a>. But it is very big and there is no way you can read it fast. Also there are no... | Peter Heinig | 108,556 | <p>Since both the following references appeared significantly later than the OP, it seems useful to add: </p>
<ul>
<li><p><a href="http://bookstore.ams.org/gsm-170/" rel="nofollow noreferrer">Donald Yau: <em>Colored Operads</em>. AMS. Graduate Studies in Mathematics
Volume 170</a></p></li>
<li><p>The review of the abo... |
2,473,220 | <p>From how I understood the question and judging from solutions I've been provided with (see graph below),</p>
<p><a href="https://i.stack.imgur.com/73RU3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/73RU3.png" alt="enter image description here"></a></p>
<p>$f(x)$ starts from an x-position, whi... | moqui | 988,087 | <p>Notice that <span class="math-container">$f$</span> is defined as:</p>
<p><span class="math-container">$$f(x):=\begin{cases}n-x\quad \text{if }\ n-\frac{1}{2}\leq x<n\\
x-n\quad \text{if }\ n\leq x<n+\frac{1}{2}\end{cases}$$</span>
for every integer <span class="math-container">$n$</span>. Now,</p>
<p><span cl... |
2,049,685 | <p>If a team 1 has a probability of p of winning against team 2. What is the probability "formula" that team one will win 7 games first. </p>
<p>There are no ties and the teams play until one t am wins 7 games </p>
| lulu | 252,071 | <p>Imagine that exactly $13$ games are played out, even though it is likely that the series will have been settled prior to the last game. The advantage here is that we know that exactly one of the teams will have won $7$ or more games and that determines the winner. To finish, we remark that for Team $1$ to win the... |
1,034,335 | <p>I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" </p>
<p>I understand the following:</p>
<p>Let the sequence $a_n$ exist such that $a_n =\frac{1}{n^2}$ </p>
<p>Then $\lim_{n\to\infty} a_n=\lim_{n... | Emanuele Paolini | 59,304 | <p>You can identify a series with the sequence of its partial sums:
$$
S_n = \sum_{k=1}^n a_k.
$$
So everything you know about sequences can be applied to series, and vice-versa.</p>
<p>However dealing with series is usually more difficult because, in general, it can be very difficult to find the limit. This is due to... |
1,034,335 | <p>I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" </p>
<p>I understand the following:</p>
<p>Let the sequence $a_n$ exist such that $a_n =\frac{1}{n^2}$ </p>
<p>Then $\lim_{n\to\infty} a_n=\lim_{n... | Alasdair | 35,574 | <p>Series can be baffling things. The trouble is that the convergence of the terms tells you nothing about the convergence of the series. We know that $\lim_{n\to\infty}a_n=0$ is <em>necessary</em> for a series $\sum_{k=0}^\infty a_n$ to converge, but it is not <em>sufficient</em>.</p>
<p>For example consider the se... |
3,412,418 | <blockquote>
<p>You have been chosen to play a game involving a 6-sided die. You get
to roll the die once, see the result, and then may choose to either
stop or roll again. Your payoff is the sum of your rolls, unless this
sum is (strictly) greater than 6. If you go over <span class="math-container">$6$</span>,... | Ross Millikan | 1,827 | <p>If you have <span class="math-container">$x$</span>, you fail and get <span class="math-container">$0$</span> with probability <span class="math-container">$\frac x6$</span>, losing <span class="math-container">$x$</span>. If you succeed, you gain a number between <span class="math-container">$1$</span> and <span c... |
2,496,817 | <p>My task is to prove the above, with $m,n \in \mathbb{N}$</p>
<p>Here is what I have:</p>
<p>$7 | (100m + n) \iff (100m +n) \mod 7 = 0$</p>
<p>$\iff (100m \mod 7 + n \mod 7) \mod 7 = 0 $</p>
<p>$\iff (2m +n) \mod 7 = 0$ </p>
<p>That is where I am stuck.</p>
| Nosrati | 108,128 | <p>$$100m+n=7k$$
$$2m+n=7k-(7\times14)m$$
$$4(2m+n)=4(7k-(7\times14)m)$$
$$m+4n=4(7k-7\times14m)-7m=7\ell$$</p>
|
10,722 | <p>I notice that geometry students frequently have difficulty with representations of 3-dimensional objects in 2 dimensions. Today, we worked with physical manipulatives in order to help visualize where right triangles can occur in 3 dimensions in both pyramids and rectangular prisms (the focus is on fluency with the P... | Joseph O'Rourke | 511 | <p>This is not what you seek, because it compares two different
physical manipulatives, rather than physical vs. virtual.
But I find it interesting partly because
my own research involves studying nets of polyhedra.</p>
<blockquote>
<p>Scott, Jacqui, Anton Selvaratnam, and Lynden Rogers. "Using Bendable and Rigid M... |
4,203,906 | <p>Does there exist real numbers a and b such that</p>
<p>(i) <span class="math-container">$a+b$</span> is rational and <span class="math-container">$a^
n +b^
n$</span>
is irrational for each natural <span class="math-container">$n ≥ 2$</span>;</p>
<p>(ii) <span class="math-container">$a+b$</span> is irrational and <sp... | zwim | 399,263 | <h4 id="study-case-i-y6a2">Study case (i)</h4>
<p>Let <span class="math-container">$s=a+b$</span> and <span class="math-container">$p=ab$</span> then <span class="math-container">$a,b$</span> are solutions of <span class="math-container">$x^2-sx+p=0$</span>.</p>
<p>Regarding this as the characteristic equation of a lin... |
73,410 | <p>Gromov proved that if $$
f,g:\left[ {a,b} \right] \to R
$$
are integrable functions, such that the function $$
t \to \frac{{f\left( t \right)}}
{{g\left( t \right)}}
$$
is also integrable, and decreasing. Then the function $$
r \to \frac{{\int\limits_a^r {f\left( t \right)dt} }}
{{\int\limits_a^r {g\left( t ... | robjohn | 13,854 | <p><em>Thanks to Mariano Suárez-Alvarez for pointing out a bad assumption I made in my previous attempt</em></p>
<p>For all $u\le v$, in $[a,b]$ we have
$$
\frac{f(u)}{g(u)}\ge\frac{f(v)}{g(v)}
$$
Assuming that $g$ is either non-negative or non-positive on all of [a,b], we get
$$
f(u)g(v)\ge f(v)g(u)
$$
Let $r\le s$. ... |
73,410 | <p>Gromov proved that if $$
f,g:\left[ {a,b} \right] \to R
$$
are integrable functions, such that the function $$
t \to \frac{{f\left( t \right)}}
{{g\left( t \right)}}
$$
is also integrable, and decreasing. Then the function $$
r \to \frac{{\int\limits_a^r {f\left( t \right)dt} }}
{{\int\limits_a^r {g\left( t ... | zyx | 14,120 | <p>The geometric interpretation of the result is fairly clear if you draw the picture of a particle with velocity vector $(f(t), g(t))$ that at time $t=a$ is at $(0,0) \quad$ (assume $g(t) > 0$ so that the particle moves to the right at all times). Decreasing $f(t)/g(t)$ means the path of the particle is convex, cu... |
1,440,522 | <p>For a function $f:[0,1]\to \mathbb{R}$, let $C$ be the set of points where $f$ is
continuous. Prove that $C$ is in the Borel $\sigma$-algebra.</p>
<p>I know that for $A=\{f(x): f(x)<a\}$ is open for each real number a, and since openness is preserved by continuity, the set $f^{-1}(A)\cap C$ should also be op... | recmath | 102,337 | <p>We can actually show that these points are a $G_{\delta}$ set (countable intersection of open sets). </p>
<p>Let </p>
<p>$$A_n=\{t \ \mathrm{s.t} \ \ \exists \delta_t>0 \ \mathrm{with} \ |f(y)-f(x)|< \frac{1}{n} \ \mathrm{when} \ x,y \in (t-\delta_t, t+\delta_t)\}$$</p>
<p>Each of these $A_n$ are open. T... |
384,006 | <p>Just came across the following question:</p>
<blockquote>
<p>Let $S=\{2,5,13\}$. Notice that $S$ satisfies the following property: for any $a,b \in S$ and $a \neq b$, $ab-1$ is a perfect square. Show that for any positive integer $d \not\in S$, $S \cup \{d\}$ does not satify the above property.</p>
</blockquote>
... | duje | 82,393 | <p>By the paper A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52. (see <a href="http://web.math.pmf.unizg.hr/~duje/pdf/dioeul2.pdf" rel="nofollow">Theorem 1b</a>),
there does not exist 4-element set the considered property and with all elements grea... |
1,998,244 | <p>Given the equation of a damped pendulum:</p>
<p>$$\frac{d^2\theta}{dt^2}+\frac{1}{2}\left(\frac{d\theta}{dt}\right)^2+\sin\theta=0$$</p>
<p>with the pendulum starting with $0$ velocity, apparently we can derive:</p>
<p>$$\frac{dt}{d\theta}=\frac{1}{\sqrt{\sqrt2\left[\cos\left(\frac{\pi}{4}+\theta\right)-e^{-(\the... | fleablood | 280,126 | <p>There are 4 groups:</p>
<p>$A$ = no siblings</p>
<p>$B$ = only brothers</p>
<p>$C$ = only sisters</p>
<p>$D$ = both brothers and sisters.</p>
<p>$A = 5$.</p>
<p>$D = 6$</p>
<p>$17 = B + D$</p>
<p>$18 = C + D$</p>
<p>So $B = 17 - D =17-6 = 11$. $C = 18 -D = 18-6 =12$</p>
<p>So there are <em>exactly</em> (n... |
3,745,273 | <p>I am looking for a way to solve :</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{x\sin(3x)}{x^4+1}\,dx $$</span></p>
<p>without making use of complex integration.</p>
<p>What I tried was making use of integration by parts, but that didn't reach any conclusive result. (i.e. I integrated <span cla... | Riemann'sPointyNose | 794,524 | <p>@Nanayajitzuki has given you a very nice solution to this problem using Leibniz' integral rule (or Feynman trick if you are a Physicist!) Really, this integral is ridiculously difficult without Complex Analysis. It's doable... but any real method is going to be highly non-trivial.</p>
<p>For another solution, we cou... |
2,994,900 | <p>Prove that <span class="math-container">$$\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1},$$</span>
where <span class="math-container">$\mu$</span> is Möbius function, <span class="math-container">$\phi$</span> is Euler's totient function, and <span class="math-container">$q$<... | arithmetic1 | 558,611 | <p>I find a paper "On some identities in multiplicative number theory", Olivier Bordellès and Benoit Cloitre, arXiv:1804.05332v2 <a href="https://arxiv.org/abs/1804.05332v2" rel="nofollow noreferrer">https://arxiv.org/abs/1804.05332v2</a></p>
<p>Using Dirichlet convolution
<span class="math-container">\begin{eqnarray*... |
50,362 | <p>I have a question about the basic idea of singular homology. My question is best expressed in context, so consider the 1-dimensional homology group of the real line $H_1(\mathbb{R})$. This group is zero because the real line is homotopy equivalent to a point. The chain group $C_1(\mathbb{R})$ contains all finite ... | Matt E | 221 | <p>Your intuition is correct, I think. I also had the experience, when first learning this material, of wanting to understand homologies explicitly in the way that you are trying to, so I encourage you to pursue your attempt to match intuition with formal definitions.</p>
<p>The basic problem you observed is that oft... |
2,461,506 | <p>I am trying to derive / prove the fourth order accurate formula for the second derivative:</p>
<p>$f''(x) = \frac{-f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x -2h)}{12h^2}$.</p>
<p>I know that in order to do this I need to take some linear combination for the Taylor expansions of $f(x + 2h)$, $f(x + h)$, $f... | Donald Splutterwit | 404,247 | <p>Exactly as Gammatester says, Taylor expand the terms upto order $4$ and verify.
\begin{eqnarray*}
-f(x+2h) &=& -f(x) &-& 2h f'(x) &-& 2h^2 f''(x) &-& \frac{4}{3} h^3 f'''(x) &-& \frac{2}{3} h^4 f''''(x) &+& O(h^5) \\
16f(x+h) &=& 16 f(x)&+& 16h f'(x... |
339,142 | <p>I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to
<a href="http://www.eng.auburn.edu/users/marghdb/MECH2110/c1_2110.pdf">this</a>,
sense is specified by two points on a line parallel to a vector. Orientation is specified by the relationship between the ve... | Christian Blatter | 1,303 | <p>For the purposes of this answer two nonzero vectors ${\bf x}$, ${\bf y}\in{\mathbb R}^d$ are considered as <em>equivalent</em> if there is a $\lambda>0$ such that ${\bf y}=\lambda\,{\bf x}$. An equivalence class is called a <em>direction</em>, and two vectors belonging to the same equivalence class are said to <e... |
350,747 | <p>Base case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds.
Let $n=k>=1$ and let $2k+1$ be true. We want to show $2(k+1)+1$ is true. So, $2(k+1)+1=(2k+2) +1$....</p>
<p>Now I'm stuck. Any ideas?</p>
| Community | -1 | <p>Let $a$ be the starting number. Then the $2n+1$ consecutive numbers are
$$a,a+1,a+2,\ldots,a+2n$$
The sum of these number is
$$S(a,n) = (2n+1)a + \dfrac{2n(2n+1)}2 = (2n+1)(a+n)$$
Clearly, $(2n+1) \mid S(a,n)$.</p>
|
2,928,196 | <p>I thought that I could take all points with rational coordinates, but this space is not discrete</p>
| bangs | 520,024 | <p>For each <span class="math-container">$n\in\mathbb{N}$</span>, let <span class="math-container">$$D_n=\{(k/2^n, 1/n)\in \mathbb{R}^2: k\in \mathbb{Z}\}.$$</span> Let <span class="math-container">$D=\cup_{n=1}^\infty D_n$</span>. Then <span class="math-container">$D$</span> is discrete. To see this, note that if <sp... |
2,476,865 | <p>As the title suggests, I'm trying to establish a good bound on</p>
<p>\begin{equation}
S(n) = \sum_{k = 2}^n (en)^k k^{-Cn/\log{n} - k - 1/2},
\end{equation}</p>
<p>where $C$ is some reasonably large positive constant. In particular I would like to have $S(n) = o(1)$, i.e., </p>
<p>\begin{equation}
\lim_{n \t... | user480881 | 480,881 | <p>This seems to do it:</p>
<p>To argue that the second term $s_2$ is indeed largest consider the ratio</p>
<p>\begin{equation}
r(d) = \frac{s_2}{s_{2 + d}}
\end{equation}</p>
<p>for some $d \leq n - 2$. Our goal is to establish $r(d) \geq 1$ for $d \geq 1$. We ignore the $-1/2$ contribution in the exponent and fi... |
1,955,591 | <p>I have to prove that ' (p ⊃ q) ∨ ( q ⊃ p) ' is a tautology.I have to start by giving assumptions like a1 ⇒ p ⊃ q and then proceed by eliminating my assumptions and at the end i should have something like ⇒(p ⊃ q) ∨ ( q ⊃ p) but could not figure out how to start.</p>
| DanielV | 97,045 | <p>If you are allowed to use the law of the exluded middle, then propositional logic proofs are straight forward. You have 2 variables, that means you have 4 cases. Then you combine the cases using LEE and Or-Elimination. It looks like:</p>
<p>Dedution 1:
$$\begin{array} {r|l}
(11) & p \land q \\
(12) & p \... |
140,500 | <p>The diagonals of a rectangle are both 10 and intersect at (0,0). Calculate the area of this rectangle, knowing that all of its vertices belong to the curve $y=\frac{12}{x}$.</p>
<p>At first I thought it would be easy - a rectanlge with vertices of (-a, b), (a, b), (-a, -b) and (a, -b). However, as I spotted no ment... | Isaac | 72 | <p>As in J.M.'s comment, the diagonals of a rectangle (any parallelogram, in fact) bisect each other, so we're looking for points on $y=\frac{12}{x}$ that are a distance of $5$ from the origin. That is, we want solutions to the system $$\left\{\begin{matrix}
y=\frac{12}{x}\\
x^2+y^2=5^2
\end{matrix}\right..$$ By sub... |
1,297,863 | <p>Is it possible to write the following function:
$$
f(x) = \begin{cases}
\frac{x-\sin x}{1- \cos x}& x\neq 0\\
0 & x=0
\end{cases}
$$
as a composition of elementary functions (including $\mathrm{sinc} (x) = (\sin x) / x)$ so that I get not large numerical errors for $x$ close to zero?</p>
<p>This is the ... | zoli | 203,663 | <p>If you use the first two members of the Taylor series of the numerator and the denominator then you get</p>
<p>$$\frac{x-\sin x}{1- \cos x}\approx \frac{x}{3}.$$</p>
<p>The error of this approximation is less than $10^{-8}$ over the interval $(-0.01,0.01).$</p>
|
606,431 | <p>Can someone explain to me how to solve this using inverse trig and trig sub?</p>
<p>$$\int\frac{x^3}{\sqrt{1+x^2}}\, dx$$</p>
<p>Thank you. </p>
| Farshad Nahangi | 50,728 | <p>You can also use integration by part: let $u=x^2$ and $dv=\frac{x}{\sqrt{1+x^2}}$ then you will have:
\begin{align*}
\int udv&=uv-\int vdu\\
&=x^2\sqrt{1+x^2}-\int2x\sqrt{1+x^2}\,dx\\
&=x^2\sqrt{1+x^2}-\frac{2}{3}(1+x^2)^{\frac{3}{2}}+C
\end{align*}
where the last integral was solved by substitution $\ u... |
2,839,945 | <blockquote>
<p>Let $p$ be a prime in $\mathbb{Z}$ of the form $4n + 1, n \in \mathbb{N}$. Show that $\left(\frac{-1}{p}\right) = 1$ (here $\left(\frac{\#}{p}\right)$ is the Legendre symbol). Hence prove that $p$ is not a prime in the ring $\mathbb{Z}[i]$.</p>
</blockquote>
<p>Here is my solution:</p>
<p>Since $p &... | Don | 571,059 | <p>For the first part, I understand that you are using the supplementary laws of quadratic reciprocity, and of course, the result is immediate. However, you can also solve the problem without that theorem. With the same notation as in your statement:</p>
<p>$\mathbb{F}_p^*$ is cyclic, so there exists $x \in \mathbb{Z}... |
96,211 | <p>A modulus of continuity for a function $f$ is a continuous increasing function $\alpha$ such that $\alpha(0) = 0$ and $|f(x) - f(y)| < \alpha(|x-y|)$ for all $x$ and $y$. I am trying to prove that an equicontinuous family $\mathcal F$ of functions has a common modulus of continuity. This seems intuitively obvious... | Alex Becker | 8,173 | <p>Building on Beni's answer, suppose that this is not right-continuous, i.e. we have some $\delta$ such that $$\sup\{|f(x)-f(y)| : d(x,y)\leq \delta+\epsilon, f\in \mathcal{F}\}-\sup\{|f(x)-f(y)| : d(x,y)\leq \delta, f\in \mathcal{F}\}>z$$
for some fixed $z>0$ and for arbitrarily small $\epsilon>0$. Then for ... |
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> Note that <span class="math-container">$\ 3\!\cdot\!5\!\cdot\!7\mid \overbrace{3\!\cdot\! 5\, (\color{#c00}{k^7\!-\!k})+ 3\!\cdot\! 7\, (\color{#c00}{k^5\!-\!k})- 5\!\cdot\! 7 (\color{#c00}{k^3\!-\!k})+ 3\!\cdot\! 5\cdot\! 7\, k^3}^{\Large{\rm sum\ = \ th... |
660,461 | <p>$A = \{a,b,c,d,e\}$</p>
<p>$B = \{a,b,c\}$</p>
<p>$C = \{0,1,2,3,4,5,6\}$</p>
<p>The first few iterations are as follows:</p>
<p>$1.$ $a,a,0$</p>
<p>$2.$ $b,b,1$</p>
<p>$3.$ $c,c,2$</p>
<p>$4.$ $d,a,4$</p>
<p>$5.$ $e,b,5$</p>
<p>$...$</p>
<p>I'm trying to figure out at which iterations we will have $x,y,z$... | coffeemath | 30,316 | <p>The least common multiple of the sizes 3,5 of sets $A,B$ is 15, so that $x=y$ will occur at any position $15k+1,15k+2,15k+5.$ For each of these, set it equal to $5$ mod 7 and solve. </p>
<p>EDIT: They should be set to 6 mod 7 since you start at step 1 with a 0 in column 3.
Thanks to @Casteels for pointing out this ... |
1,483,802 | <p>Take $B(0,1)$ the ball in $\mathbb{R}^2$ with the normalized Lebesgue measure $\lambda$ such that $\int_{B(0,1)} d \lambda=1.$</p>
<p>Now, I want to show, or give a counterexample that this is false, that for all $f \in H^1_0(B(0,1))$ we have
for fixed constants $a,b>0$ and any(!) $p \in (2,\infty)$
\begin{equat... | Xiao | 131,137 | <p>Can you use Sobolev embedding theorem?</p>
<p>Given your domain is a ball, $W_0^{1,2}(B) = H_0^{1}(B)$.
In this case $W^{1,p}$ and $\mathbb{R}^n$ where $p=n=2$, we have the continuous embedding
$$W^{1,2}_0(B) \hookrightarrow L^q(B) \text{ for all } q\in [1,+\infty),$$
that is
$$\|f\|_q \leq C \bigg(\int_B |f|^2 d... |
2,554,448 | <p>Beside using l'Hospital 10 times to get
$$\lim_{x\to 0} \frac{x(\cosh x - \cos x)}{\sinh x - \sin x} = 3$$ and lots of headaches, what are some elegant ways to calculate the limit?</p>
<p>I've tried to write the functions as powers of $e$ or as power series, but I don't see anything which could lead me to the righ... | celtschk | 34,930 | <p>Using power series:
$$\begin{aligned}
\frac{x(\cosh x-\cos x)}{\sinh x-\sin x}
&= \frac{x\left((1+\tfrac12 x^2 + O(x^4)) - (1-\tfrac12 x^2 + O(x^4)\right)}
{(x+\frac16 x^3 + O(x^5)) - (x - \frac16 x^3 + O(x^5))}\\
&= \frac{x\left(x^2 + O(x^4)\right)}
{\frac13 x^3 + O(x^5)}\\
&= \frac{1 + O(x^2)}{\tfrac13... |
525,326 | <p>If all elements of $S$ are irrational and bounded from below by $\sqrt 2$ then $\inf S$ must be irrational .</p>
<p>I would say this statement is true since $S=\{ \sqrt 2, \sqrt 3, \sqrt 5,\ldots\}$ the greatest lower bound is $\sqrt 2$ which is irrational and bounded from below the sequence. </p>
<p>Is this corre... | Brian M. Scott | 12,042 | <p>HINT: All elements of $S=[2,3]\setminus\Bbb Q$ are irrational and bounded below by $\sqrt2$, but $\inf S>\sqrt2$; what is $\inf S$?</p>
|
2,184,776 | <p>So there's an almost exact question like this here: </p>
<p><a href="https://math.stackexchange.com/questions/576268/use-a-factorial-argument-to-show-that-c2n-n1c2n-n-frac12c2n2-n1#576280">Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$</a></p>
<p>However, I'm getting stuck in just... | Icycarus | 409,911 | <p>Prove by the combinatoric way:</p>
<p>You can actually simplify this equation to be $C(2n+2,n+1)=2C(2n,n+1)+C(2n,n)$</p>
<p>Now, on the left-hand side, we assume we have $2n+2$ elements, and we want to pick $n+1$ elements. That would give us $C(2n+2,n+1)$</p>
<p>On the right-hand side, this is one way we can pick... |
32,021 | <p>On more than one occasion, always with an explicit disclaimer, I
have posted a comment of more than 600 characters as an "answer".
I have done this because I have quite often seen other people do it, and
I have never once, in 5 years in Maths.SE, seen anyone object to the
practice. But a comment I posted i... | Xander Henderson | 468,350 | <p><strong>The answer box is meant for answers, not comments.</strong></p>
<p>The Stack Exchange model is meant to facilitate the construction of a high-quality database of questions and authoritative answers. Stack Exchange is not a <a href="https://meta.stackexchange.com/questions/65261/is-stack-overflow-a-social-ne... |
32,021 | <p>On more than one occasion, always with an explicit disclaimer, I
have posted a comment of more than 600 characters as an "answer".
I have done this because I have quite often seen other people do it, and
I have never once, in 5 years in Maths.SE, seen anyone object to the
practice. But a comment I posted i... | user1729 | 10,513 | <p>In the spirit of the question: This started life as more of an extended comment than an answer. I think it has morphed into an answer now though.</p>
<p>Math.SE seems to be based around the idea that every problem is soluble by a single person. This is clearly not the case, as, for example, in research mathematics s... |
2,792,770 | <p>I found the following question in a test paper:</p>
<blockquote>
<p>Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say
about $a$?</p>
</blockquote>
<p>Monoids are associative and have an identity element. Semigroups are just associative. </p>
<p>I'm not sure what we can say about $a... | wayne | 557,397 | <p>Take $\Omega = \{0,1\}$, $\mathcal{F} = \{\emptyset,\Omega\}$, $\mathbb{P}(\emptyset) = 0$, $\mathbb{P}(\Omega) = 1$, $\Omega'=\Omega$, $\mathcal{F}' = 2^{\Omega}$, and $X(\omega) = \omega$ for every $\omega \in \Omega$. Since $\{1\} \in \mathcal{F}'$ and $X^{-1}(\{1\}) = \{1\} \notin \mathcal{F}$, $X$ is not $\math... |
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