qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,225 | <p>If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant.</p>
<p>My question is suppose $f$ is continuous and it satisfies $f(x)=f(2x+1)$, then can the domain of $f$ be restricted so that $f$ doesn't remain a constant. If yes, then ... | Community | -1 | <p>As in <a href="https://mathoverflow.net/questions/31990/continuous-functions-remaining-constant">the previous proof</a> of $f$ being constant on $\mathbb{R}$, define $g(x) = f(x-1)$, so that $g(x) = g(2x)$; the domains of $f$ and $g$ are just shifted versions of each other.</p>
<p>Certainly, if the domain of $g$ is... |
1,278,848 | <p>Based on <a href="https://math.stackexchange.com/questions/1267021/let-m-subseteq-mathbbrk-manifold-topology-vs-trace-topology/1267760?noredirect=1#comment2573732_1267760">this</a> question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the to... | Tim Raczkowski | 192,581 | <p>The Riemann sphere is a compact complex manifold without boundary.</p>
|
1,278,848 | <p>Based on <a href="https://math.stackexchange.com/questions/1267021/let-m-subseteq-mathbbrk-manifold-topology-vs-trace-topology/1267760?noredirect=1#comment2573732_1267760">this</a> question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the to... | aGer | 191,887 | <p>Well there are several examples. @Dorebell mentioned one example.
Here are some other compact manifolds without boundary:</p>
<ul>
<li>Torus</li>
<li>Double Torus</li>
<li>Klein bottle</li>
</ul>
|
1,278,848 | <p>Based on <a href="https://math.stackexchange.com/questions/1267021/let-m-subseteq-mathbbrk-manifold-topology-vs-trace-topology/1267760?noredirect=1#comment2573732_1267760">this</a> question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the to... | Andrew D. Hwang | 86,418 | <p>$\newcommand{\Reals}{\mathbf{R}}$Comparing this question with your linked question, the central issue seems to be the term "closedness", which perhaps feels bothersome because manifolds are unions of open sets.</p>
<p>If that's really the question, the resolution comes down to "relative topology", how "open" and "c... |
297,812 | <p>If $a-b=b-c$ .How to find the value of $a^2-2b^2+c^2$</p>
| Inquest | 35,001 | <p>\begin{align}
a-b&=b-c\\
a+c&=2b\\
(a+c)^2&=(2b)^2\\
\end{align}</p>
<blockquote class="spoiler">
<p>\begin{align}a^2+c^2+2ac&=4b^2\\a^2+c^2-2b^2&=2b^2-2ac\\\end{align}</p>
</blockquote>
|
1,029,868 | <p>Let $$
A=\begin{bmatrix}
1 & 1 & 2\\
1 & 2 & 1\\
2 & 1 & 1
\end{bmatrix}$$</p>
<p>Show that $ A^-=\dfrac{1}{4}(-A^2+4A+I)$</p>
<p>I have absolutely no clue how to do this. Could someone be kind enough to explain and provide and answer? I believe it has something to do with the Cayley-Hamili... | TenaliRaman | 29,755 | <ul>
<li>Show that $|\lambda I_3 - A| = \lambda^3 - 4 \lambda^2 - \lambda + 4$</li>
<li>Use Cayley Hamilton Theorem, to show that $A^3 - 4 A^2 - A + 4I_3 = 0$</li>
<li>Show that A inverse exists (how?)</li>
<li>Multiply the equation from step 2 with A inverse and rearrange</li>
</ul>
|
21,857 | <p>This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the <a href="http://math.arizona.edu/~swc/aws/07/speakers/index.html">notes of Conrad</a>. </p>
<p>Reading Conrad's notes (and e.g. those of Bosch) it struck me that the prime spec... | Kevin Buzzard | 1,384 | <p>I am surprised that Brian got to this one first without making what I thought was another obvious comment: affinoids are Jacobson rings! A function which is zero at all points of an affinoid rigid space corresponds to an element of your affinoid algebra which is in all maximal ideals and hence (by Jacobson-ness) is ... |
21,857 | <p>This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the <a href="http://math.arizona.edu/~swc/aws/07/speakers/index.html">notes of Conrad</a>. </p>
<p>Reading Conrad's notes (and e.g. those of Bosch) it struck me that the prime spec... | Emerton | 2,874 | <p>Another point to bear in mind, in addition to those raised by Brian and Kevin, is that generic points (in the sense of non-maximal prime ideals) don't make sense in analytic geomtery.</p>
<p>For example, the Tate algebra $\mathbb Q_p\langle\langle x\rangle \rangle$ contains
one non-maximal prime ideal, the zero id... |
1,322,076 | <p>Hey can anybody help me with the following proof? I am trying to solve the following limit using epsilon delta and I have found the limit to be 1/3 using the squeeze theorem and have got to this thus far but am a bit confused where I go now as I have both a 3x and a sinx when trying to find an epsilon??
Thanks in ad... | xanthousphoenix | 209,166 | <p>You are using the wrong definition of limits for this case. When dealing with limits at infinity, you want to use <a href="http://www.millersville.edu/~bikenaga/math-proof/limits-at-infinity/limits-at-infinity.html" rel="nofollow">this</a> definition. As RowanS stated, you can use the fact that the absolute value of... |
2,010,255 | <p>While finding the Taylor Series of a function, <strong>when</strong> are you allowed to substitute? And <strong>why</strong>?</p>
<p>For example:</p>
<p>Around $x=0$ for $e^{2x}$ I apparently am allowed to substitute $u=2x$ and then use the known series for $e^u$. But for $e^{x+1}$ I am not allowed to substitute $... | Enrico M. | 266,764 | <p>The quantity $2x$ is a product and as $x\to 0$ it remains a small number.</p>
<p>The quantity $x+n$ for $n\neq 0$ is not a little quantity anymore, and so you are not anymore around zero but you're around $n$.</p>
|
92,867 | <p>Suppose we have some random variable $X$ that ranges over some sample space $S$. We also have two probability models $F$ and $G$. Let $f(x)$ and $g(x)$ be the probability density functions for these distributions. Does the following quantity $$ \log \frac{f(x)}{g(x)} = \log \frac{P(F|x)}{P(G|x)}- \log \frac{P(F)}{P(... | Elvis | 21,435 | <p>I am totally confused by the last comment made by Michael (the answer is ok, it is the link with logistic regression which went too far for me). Logistic regression is to be used when you have pairs of observations (X, Y) where Y is a binary variable (taking values in {0,1}) which is modeled as a Bernoulli variable ... |
121,909 | <p>I came across this question while studying primitive roots. I know it has something to do with the fact that if the order of $a$ is $m$ then for every $k \in \mathbb{Z}$, the order of $a^k$ is $m/(m,k)$. The question is as follows: </p>
<blockquote>
<p>Let $p$ be an odd prime. Prove that $a^2$ is never a primi... | bgins | 20,321 | <p>If $(a,p)=1$, then $1\equiv a^{p-1}=(a^2)^\frac{p-1}{2}$ implies that $\text{ord}_p(a^2)\le\frac{p-1}{2}$.</p>
|
1,237,528 | <p>$$ \displaystyle {\int_{0}^{z}} \sqrt {1 + \tan^2(\dfrac{\pi}{4} \dfrac{z}{H} )} dz $$</p>
<p>_</p>
<p>$$ gives $$ </p>
<p>_</p>
<p>$$ \dfrac{4H}{\pi} {\sinh^{-1}} ( {\tan \dfrac{\pi}{4} \dfrac{z}{H} } ) $$</p>
<p>Please advise solution</p>
<p>edit:- </p>
<p>I can get to </p>
<p>$$\dfrac{4H}{\pi} \displaysty... | Mark Viola | 218,419 | <p>There are always more than one way to represent a solution. </p>
<p>So, let's note a couple of things here.</p>
<hr>
<p>First, note that the hyperbolic sine function $\sinh x =\frac12 (e^x-e^{-x})$ has inverse function </p>
<p>$$\sinh^{-1}x=\log\left(x+\sqrt{1+x^2}\right)$$</p>
<p>To see this, let's solve the ... |
138,723 | <p>By cleaning up a notebook, I mean how can I hide all the codes in the notebook so that the end-users can't see it? I saw Eric Schulz's famous interactive calculus textbook, the users can't see the code, and there is no cell brackets on the right hand side of the CDF. </p>
| m_goldberg | 3,066 | <h3>Update</h3>
<p>I have incorporated Kuba's improvement into the code.</p>
<p>Here is how I would do it.</p>
<ol>
<li><p>In a working notebook (not the target notebook) put the following code.</p>
<pre><code>With[{nb = target},
SetOptions[nb, ShowCellBracket -> False];
SetOptions[#, CellOpen -> False] &... |
2,090,512 | <p>You can calculate the <strong>volume of a parallelepiped</strong> by $|(A \times B) \cdot C|$, where $A$, $B$ and $C$ are vectors. I wonder does the order matter? If it does how, is it determined. I know I can just put it in a matrix and calculate the determinant but I would like to know how it is in this case. </p... | David K | 139,123 | <p>If you know that
<a href="https://math.stackexchange.com/questions/314275/scalar-triple-product-why-equivalent-to-determinant">the scalar triple product is equal to the determinant of a matrix
whose rows are the components of the vectors</a>,
and if you recall the effects of operations on the rows of a matrix,
then ... |
2,762,715 | <blockquote>
<p>Let $\mathrm a,b$ are positive real numbers such that for $\mathrm a - b = 10$, then the smallest value of the constant $\mathrm k$ for which $\mathrm {\sqrt {x^2 + ax}} - {\sqrt{x^2 + bx}} < k$ for all $\mathrm x>0$, is? </p>
</blockquote>
<p>I don't get how to approach this problem. Any help ... | trancelocation | 467,003 | <p>An elementary way ($x>0$):
$$f(x) = \sqrt{x^2 + ax} - \sqrt{x^2+bx} = \frac{x^2 + ax - (x^2+bx)}{\sqrt{x^2 + ax} + \sqrt{x^2+bx}} =\frac{(a - b)x}{x\sqrt{1 + \frac{a}{x}} + x\sqrt{1 + \frac{b}{x}}}= \frac{(a - b)}{\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}} < \frac{a-b}{2}= 5$$</p>
|
3,187,451 | <p>Can you help me find a function <span class="math-container">$f(X,Y)$</span>, such that <span class="math-container">$f(1,x) = f(x,1) = f(\ln x, \ln x)$</span>?</p>
<p>Either always, for all <span class="math-container">$x$</span> or in the limit <span class="math-container">$x$</span> tends to infinity, all these ... | AsdrubalBeltran | 62,547 | <p><span class="math-container">$$f(X,Y)=(X-1)(Y-1)(X-Y)$$</span>
<span class="math-container">$f(x,1)=f(1,x)=f(\ln{x},\ln{x})=0$</span></p>
|
3,203,282 | <p>Given that <span class="math-container">$C[-\pi,\pi]$</span> is complete:
How can we prove, by using the supremum norm, that the space:</p>
<p><span class="math-container">$$C_p[-\pi,\pi]=\{f\in C[-\pi,\pi]\mid f(-\pi)=f(\pi)\}$$</span></p>
<p>is also complete? thank you!</p>
| Frank W | 552,735 | <p>Another possible way is to differentiate <span class="math-container">$\sin^2x$</span> and observe that<span class="math-container">$$[\sin^2x]'=2\sin x\cos x=\sin 2x$$</span></p>
<p>Thus, using the taylor series for <span class="math-container">$\sin x$</span> gives<span class="math-container">$$\sin 2x=\sum\limit... |
3,033,812 | <p>My problem: If there are 5 different candies in a jar and a child wants to take out one or more candies, how many ways can this be done? </p>
<p>I said it is <span class="math-container">$^5C_1 -\; ^5C_0 = 5-1 = 4$</span> ways. The <span class="math-container">$-1$</span> for the unwanted case using this trick:</p>... | Kyky | 423,726 | <p>Think of it like this: The child can either take a specific candy or not take it. This means we have <span class="math-container">$2$</span> possibilities for whether this candy is taken or not. Given we have <span class="math-container">$5$</span> candies, we have <span class="math-container">$2\cdot2\cdot2\cdot2\c... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | Ross Millikan | 1,827 | <p>The steps to the third line seem routine, trying to find the terms of the inductive hypothesis. Once you are at the third line you have to decide to split the $64$ into $7+57$. You might just notice that both numbers are important in the problem and try it. You might notice that splitting out $7$ of the second te... |
1,889,957 | <p>I'm a bit rusty on my math notations and I'd like to write that:</p>
<blockquote>
<p>It exists a unique element $z$ such that $z$ belongs to the collection of values returned by $f(x,y)$</p>
</blockquote>
<p>Honestly I'm not just rusty I'm also mostly ignorant of math except from basic functions and basic matrix... | Roby5 | 243,045 | <p>Let </p>
<p>$$P=\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}$$</p>
<p>$$Q=\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}+\frac{a}{a+b}$$</p>
<p>$$R=\frac{c}{b+c}+\frac{d}{c+d}+\frac{a}{d+a}+\frac{b}{a+b}$$</p>
<p>We have $$Q+R=4\tag{1}$$</p>
<p>$$P+Q=\frac{a+b}{b+c}+\frac{b+c}{c+d}+\frac{c+d}{d+a}+\frac{d... |
390,129 | <p>Let <span class="math-container">$O$</span> be a <span class="math-container">$d$</span>-dimensional rotation matrix (i.e., it has real entries and <span class="math-container">$OO^T = O^TO = I$</span>). Let <span class="math-container">$\mathbf{x}$</span> be a uniformly random bitstring of length <span class="math-... | Marco | 143,536 | <p>Here is an attempt to the problem for a worst-case <span class="math-container">$O$</span>, with worse constants. So fix <span class="math-container">$O$</span>, letting <span class="math-container">$o_i$</span> denote its <span class="math-container">$i$</span>th row, and take <span class="math-container">$X$</span... |
3,659,413 | <p>I'm reading the book "Topological Graph Theory" by Gross and I've gone through a fair bit of it. It seems like the entire book is leading up to being able to imbed a group onto a surface, and I have no idea why you would want to do that.
I am a physics major and not very advanced in math.
Any insight would be appre... | Jonas Linssen | 598,157 | <p>Many graph theoretic problems become easy when it is known that the graph is planar ie. can be embedded in the sphere. For example the isomorphism problem for planar graphs can be solved in polynomial time, they can be 5colored in polynomial time (I don’t know about the time complexity of finding a 4coloring though)... |
2,900 | <p>I saved an <code>InterpolationFunction</code> in a ".mx" files using <code>DumpSave</code> on a variable that was scoped by a <code>Module</code>. Here is a stripped-down example:</p>
<pre><code>Module[{interpolation},
interpolation=Interpolation[Range[10]];
DumpSave["interpolation.mx", interpolation];
]
</co... | JxB | 63 | <p>Here is another method, although I don't know how to capture the symbol name programatically...</p>
<pre><code>On[General::newsym];
Get["/tmp/test.mx"];
Off[General::newsym];
(* General::newsym: Symbol a$1772 is new. >> *)
</code></pre>
|
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | David G. Stork | 210,401 | <p>Perhaps this graph will help reveal the answers (abscissa in radians):</p>
<p><a href="https://i.stack.imgur.com/UDvZW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UDvZW.png" alt="enter image description here" /></a></p>
|
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | g.kov | 122,782 | <blockquote>
<p>Find the solutions to the equation<br />
<span class="math-container">\begin{align} 2\tan(2x)=3\cot(x),\quad 0^\circ<x<180^\circ \tag{1}\label{1}\end{align}</span></p>
</blockquote>
<p>As it was already noted, <span class="math-container">$\tan x$</span> is not defined
on the whole range <span cla... |
2,529,262 | <p>I have five real numbers $a,b,c,d,e$ and their arithmetic mean is $2$. I also know that the arithmetic mean of $a^2, b^2,c^2,d^2$, and $e^2$ is $4$. Is there a way by which I can prove that the range of $e$ (or any ONE of the numbers) is $[0,16/5]$. I ran across this problem in a book and am stuck on it. Any help w... | Michael Rozenberg | 190,319 | <p>By C-S $$(1^2+1^2+1^2+1^2)(a^2+b^2+c^2+d^2)\geq(a+b+c+d)^2$$ or</p>
<p>$$4(a^2+b^2+c^2+d^2)\geq(a+b+c+d)^2$$ or</p>
<p>$$4(20-e^2)\geq(10-e)^2$$ or
$$(e-2)^2\leq0$$ or
$$e=2.$$</p>
<p>This method works in the general case.</p>
<p>Given: $a+b+c+d+e=k$ and $a^2+b^2+c^2+d^2+e^2=l$.</p>
<p>Find the range of $e$. </... |
3,258,249 | <p><span class="math-container">$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$</span></p>
<p>It's clear that we can simplify the limit a little bit, after which we get:</p>
<p><span class="math-container">$\lim\limits_{n\to\infty}{(\frac{1}{4})^{n}\sum\limits_{... | user10354138 | 592,552 | <p><strong>Hint for a probabilistic proof</strong>: look at the negative binomial distribution as the sum of independent geometric distributions, and apply the central limit theorem.</p>
|
77,504 | <p>I'm in the embarrassing situation that I want to ask a question that
was <a href="https://mathoverflow.net/questions/14175/how-to-learn-about-shimura-varieties">already asked</a>, but (for complicated reasons) never answered. I'd
like to try with a blank slate.</p>
<p>Shimura varieties show connections to a lot of... | Andrei Halanay | 1,220 | <p>I think a great introduction to this subject is given in two articles of J.S. Milne: one from the book
<a href="http://www.claymath.org/library/proceedings/cmip04c.pdf">James Arthur, David Ellwood, Robert Kottwitz (eds.)-Harmonic Analysis, the Trace Formula, and Shimura Varieties</a> and a shorter <a href="http://ww... |
3,087,933 | <p>I read in the book <em>A First Course in Probability</em> by Sheldon Ross the following statement:</p>
<blockquote>
<p><strong>Technical Remark.</strong> We have supposed that <span class="math-container">$P(E)$</span> is defined for all the events <span class="math-container">$E$</span> of the sample space. Actu... | BadAtAlgebra | 611,990 | <p>The answer is that there is <span class="math-container">$0$</span> chance that a randomly chosen number is even, because the set of real numbers is much larger than the set of even numbers (a bigger type of infinity, to be precise).</p>
|
65,270 | <p>On <a href="https://crypto.stanford.edu/pbc/notes/elliptic/divisor.html" rel="nofollow noreferrer">this page</a>, the author states:</p>
<blockquote>
<p>It turns out this definition can be extended to points of order 2, and also the point O (when we homogenize the functions and work over the projective plane). Moreo... | Gooz | 16,218 | <p>Given a rational function $f$ on an elliptic curve (or a smooth projective curve) $E$, we have that $$\sum_{x\in E} \textrm{ord}_x(f) =0. $$ If $\textrm{ord}_x(f) >0$, we say that $f$ has a zero of order $\textrm{ord}_x(f)$ at $x$. If $\textrm{ord}_x(f)<0$, we say that $f$ has a pole of order $-\textrm{ord}_x(... |
4,122,425 | <p>Let’s say a corona test is correct with <code>p=0.8</code>. If I now take two tests. What’s the probability that I get a correct result?</p>
<p>I think thought of <code>0.8*0.8</code>, but that makes now sense, since it should not decrease and <code>0.8+0.8</code> gives a probability over 1, which makes no sense eit... | Garo | 526,127 | <ul>
<li>If <code>0.8</code> would be the probability of a correct one then <code>1 - 0.8 = 0.2</code> would be the probability of a incorrect one</li>
<li><code>0.8 * 0.8 = 0.64</code> will be the probability that they are <strong>both</strong> correct</li>
<li>Which means that the reverse: <code>1 - 0.64 = 0.36</code... |
2,807,611 | <p>I know the answer is $n=6$, but can't figure out how to solve.
I tried dividing by $n!$, but didn't work because there isn't one in RHS to simplify... also tried using Gamma function properties, but didn't work either... </p>
<p>Any help would be appreciated.</p>
<p>Thanks.</p>
| SlipEternal | 156,808 | <p>Multiply both sides by 5!. That gives you:</p>
<p>$n!((n+1)(n+2)-1) = 330\cdot 5!$</p>
<p>So, you now have the general format of a solution. We know that $n!$ divides $330\cdot 5!$, so $n\le 6$. Trial and error will get you there quickly.</p>
<p>$1!(2\cdot 3-1) = 5\cdot 1! \neq 330\cdot 5!$</p>
<p>$2!(3\cdot 4-... |
5,231 | <p>I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.</p>
<p>How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?</p>
<p>I've tried to get the plane rotation from the plane's normal:</p>
<pre>... | Community | -1 | <p>Try to represent rotations using matrices instead of angles - then finding the inverse is easy.</p>
|
1,397,160 | <p>How do I prove that in a finite group G, for each element in G there is natural power (say $k$) which depends on g,such that $g^k=e$ ?
I need to show the existence and the dependence on which $g$ I choose.</p>
<p>I tried write it that way, but I don't have any direction in the proof: </p>
<p>$$G\:=\:\left|n\right|... | Chinny84 | 92,628 | <p>Having had to search "Coefficient of area expansion" (and I did physics at uni) you did not explain that you are working with this
$$
L = L_0\left(1+\alpha\Delta T\right)
$$
so we have
$$
A = L^2 = L_0^2\left(1+2\alpha\Delta T + \alpha^2(\Delta T)^2\right)\approx L_0^2\left(1+2\alpha\Delta T\right)
$$
you ignore te... |
1,498,048 | <p>I can´t prove this problem. Can you help me? The problem says:</p>
<p><em>If $\{X_n\}$ is a sequence of identically distributed random variables with finite mean, then
$$lim_{n\to\infty}\frac{1}{n}\mathbb{E}\Big[\max_{1\leq j\leq n} |Xj|\Big] = 0$$
[HINT: Use Exercise 17 to express the mean of the maximum.]</em></p... | aghil alaee | 492,049 | <p>since h(x)=|f(x)| for any real valued function f, is a convex function(why?) then max(|Xi|) is also convex. by using Jensen's inequality you can prove this. be happy</p>
|
1,822,562 | <p>Please explain what method you used to prove so.
$$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$</p>
| ncmathsadist | 4,154 | <p>Note that $\tan(x)\sim x$ as $x\to 0$. </p>
|
3,338,388 | <p>I tried to calculate the expression:
<span class="math-container">$$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right)$$</span>
in Wolframalpha, but it does not interpret it correctly. </p>
<p>Could someone help me type it in and get the ans... | bilgamish | 558,586 | <p>Here is the screenshot of MS Excel spreadsheet:</p>
<p><a href="https://i.stack.imgur.com/d4FVG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d4FVG.png" alt="enter image description here"></a></p>
|
138,079 | <p>I want to find an elegant method to rearrange these two sublists:</p>
<pre><code>SeedRandom[1]
list = {RandomInteger[10, {4, 2}], RandomInteger[{10, 30}, {4, 2}]}
</code></pre>
<blockquote>
<p>{{{1,4},{0,7},{0,0},{8,6}},{{11,20},{11,11},{25,17},{27,16}}}</p>
</blockquote>
<p>Make these two sublists’ elements ha... | WReach | 142 | <p>Here is a shorter, though not necessarily prettier, solution:</p>
<pre><code>ReplaceList[list, {{___, l_, ___}, {___, r_, ___}} :> {l, r} -> N@EuclideanDistance[l, r]] //
SortBy[Last] //
DeleteDuplicates[#, #[[1, 1]] == #2[[1, 1]] || #[[1, 2]] == #2[[1, 2]]&] & //
{#[[All, 1, 1]] // Reverse, #[[All, 1... |
138,079 | <p>I want to find an elegant method to rearrange these two sublists:</p>
<pre><code>SeedRandom[1]
list = {RandomInteger[10, {4, 2}], RandomInteger[{10, 30}, {4, 2}]}
</code></pre>
<blockquote>
<p>{{{1,4},{0,7},{0,0},{8,6}},{{11,20},{11,11},{25,17},{27,16}}}</p>
</blockquote>
<p>Make these two sublists’ elements ha... | Jack LaVigne | 10,917 | <p><strong><em>Elegant</em></strong> is completely subjective. It could mean <strong><em>short</em></strong> or possibly <strong><em>easy to follow</em></strong> or something entirely different.</p>
<p>The solution below is not the shortest but I do find it relatively easy to follow.</p>
<pre><code>SeedRandom[1]
list... |
3,971,025 | <p>I need to find the number of conjugated to the permutation (12)(34) in the symmetric group <span class="math-container">$S_6$</span> of rank 6</p>
<p>My answer is 6! = 720</p>
<p>Is this correct?</p>
<p>I concluded that (12)(34)=(12)(34)(5)(6) and the number of combinations for <span class="math-container">$S_6$</sp... | Tanner Swett | 13,524 | <p>Here's one way to "translate" it.</p>
<blockquote>
<p>If <span class="math-container">$L^+(P,N_0)$</span> is the set of functions <span class="math-container">$f:P\rightarrow N_0$</span> with a property such that
<span class="math-container">$$\exists\; n_0 \in N_0 \; \forall \; p \in P \;$$</span></p>
<p>... |
324,557 | <p>Map the common part of the disks $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. Choose the mapping sot hat the two symmetries are preserved.</p>
<p>I don't really know how to approach this??</p>
<p>Any suggestions on how to start constructing such a linear transformation??</p>
<p>Thanks in advance!... | Ittay Weiss | 30,953 | <p>To solve such questions it helps to construct small examples of transitive relations <em>in the most obvious way</em>. So, let $A=\{1,2,3\}$ and have $R\{(1,2),(2,3),(1,3)\}$. It is constructed by force to be transitive, but computing $R\circ R$ reveals a that $R\circ R\ne R$. </p>
<p>The moral is not so much any o... |
199,738 | <p>It is known that, if a function $f$ from a planar domain $D$ to a Banach space $A$ is weakly analytic [i.e. $l(f)$ is analytic for every $l$ in $A^*$], then $f$ is strongly analytic [i.e. $\lim_{h \to 0} h^{-1}[f(z+h)-f(z)]$ exists in norm for every $z$ in $D$].</p>
<p>Now the question is, if above $f$ is assumed t... | Rabee Tourky | 39,780 | <p>Regarding the clarified question with finite dimensional domain. Let $X$ be an infinite dimensional separable and reflexive Banach space. Its unit ball $B$ is weakly compact and metrizable. It is also convex. </p>
<p>So by Hahn–Mazurkiewicz theorem there exists a continuous function $f\colon [0,1]\to B$ that is o... |
187,459 | <p>What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?</p>
<p>Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one vertex in common, then can we characterize them in some way?</p>
<p>When is it possible to draw such a graph... | Brendan McKay | 9,025 | <p>The number of vertices $n$ must be even or the number of 4-cycles is not an integer. The number of simple connected quartic graphs with the first condition is 0 for $n<12$ and $2,4,25,459$ for $n=12,14,16,18$. One of those on 12 vertices is the <a href="https://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow... |
295,076 | <p>If a finite-dimensional vector space $V$ is a direct sum of two subspaces $W_1$ and $W_2$, prove that $V^* = W_1^0 \oplus W_2^0$.</p>
<p>Where $V^*$ is the dual space of $V$ and $W^0$ is the annihilator of $W$.</p>
| DonAntonio | 31,254 | <p>Hint:</p>
<p>Look at $\,V^*/W_1^0\,$ and check your last question, already answered.</p>
|
3,070,788 | <p>Can anyone explain to me why the variance of the standard normal distribution is 1? I am trying to understand the mechanism behind standardising random variable. While I know minus the variable by the mean is like shifting the graph to make it centre at the origin, I don't know why dividing it by SD makes the variab... | Community | -1 | <p>It is immediate that <span class="math-container">$K=7$</span> and <span class="math-container">$Q=9$</span>. Then <span class="math-container">$L+2Z=37$</span> and <span class="math-container">$L+Z=24$</span> yield <span class="math-container">$Z=13,L=11$</span>. <span class="math-container">$X$</span> and <span cl... |
762,472 | <p>Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$.</p>
<hr>
<p>It is one of those questions that seem very intuitive but very hard to prove for me. How do I show that there are "exactly... | ajotatxe | 132,456 | <p><strong>A proof</strong></p>
<p>Let be $s$ a tangent line through $P$. Call $T$ the point in which the line touches $C$. The radius at $T$ is perpendicular to $s$.</p>
<p>So you have to find the points $T$ such that the angle $\angle PTO$ is right (I have called $O$ the center of $C$).</p>
<p>But these points hav... |
762,472 | <p>Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$.</p>
<hr>
<p>It is one of those questions that seem very intuitive but very hard to prove for me. How do I show that there are "exactly... | colormegone | 71,645 | <p>A proof using analytic geometry --</p>
<p>We can place the circle of radius $ \ r \ $ with its center at the origin, so its equation is $ \ x^2 \ + \ y^2 \ = \ r^2 \ $ . We can also pick a point $ \ (C, 0 ) \ $ on the positive $ \ x-$ axis. (This is equivalent to just saying we have some circle and some external ... |
993,767 | <p>Suppose $V$ is an inner product space over $\mathbb F$ and $u$,$v$ ∈ $V$ and
$\|u\| ≤ \|u + av\|$
for all $a$ ∈ $\mathbb{F}$.Then I want to show that $u$ and $v$ are orthogonal.I want to prove it geometrically.Somebody please give me some hint.</p>
| mookid | 131,738 | <p>Consider a bijection $\Bbb N \to \Bbb Q$: $f(n)$, and define $a_n = f(n)$.</p>
<p>Now let $x\in \Bbb R$. There is a sequence of rationals $(q^{(x)}_n)$ such as
$$
q^{(x)}_n\uparrow x
$$</p>
<p>Define $b_1 = q_1^{(x)}$ and, for every $n$ take $b_{n+1} \in \{q_k^{(x)}: k > n \}
\cap \{ f(k): k > n \}
$. This w... |
2,847,419 | <p>I know that <br/>
$\sigma , \delta$ be 2 function then <br/>
$1)$ $\sigma \circ \delta$ is onto or one-one if both $\sigma $ and $\delta$ is onto or one one.<br/>
I can prove this fact .
I wanted to find the counterexample for both cases if the converse is not true.
<br/> Any Help will be appreciated </p>
| Mohammad Riazi-Kermani | 514,496 | <p>$$f'(x)=k(x+e^x)^{k-1} \times (1+e^x)=0 $$</p>
<p>has only one solution which is where $x+e^x=0$ and that is the point that you want to approximate. </p>
<p>The answer should be negative so $x=0.567$ is problematic. </p>
|
1,450,176 | <p>I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$.</p>
<p>I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this </p>
<p>problem : ${1}^{-\infty }$ then i can't judge if this limit equal's $1$ , </p>
<p>because it is indeterminate case ,T... | Victor | 142,550 | <p>I see you're a high school teacher so you're familiar with the following concepts :</p>
<blockquote>
<p>$\bullet$ $\sin(\frac{1}{x}) \simeq \frac{1}{x} - \frac{1}{6x^3} \text{ } [\text{as x $\rightarrow$ $\infty$}]$</p>
<p>$\bullet $ $ \lim_{x \to \infty} (1-\frac{k}{x})^x = e^{-k} $</p>
</blockquote>
<p>... |
1,450,176 | <p>I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$.</p>
<p>I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this </p>
<p>problem : ${1}^{-\infty }$ then i can't judge if this limit equal's $1$ , </p>
<p>because it is indeterminate case ,T... | egreg | 62,967 | <p>Compute the limit of the logarithm:
\begin{align}
\lim_{x\to\infty}(1-x)\log(x\sin(1/x))&=
\lim_{t\to0^+}\left(1-\frac{1}{t}\right)\log\frac{\sin t}{t}
\\[6px]
&=\lim_{t\to0^+}\log\frac{\sin t}{t}-\lim_{t\to0^+}\frac{\log\sin t-\log t}{t}\\[6px]
&=-\lim_{t\to0^+}\left(\frac{\cos t}{\sin t}-\frac{1}{t}\ri... |
94,134 | <p>I have a feeling that the following inequality should be very easy to prove:</p>
<p>$$
x^n \geq \prod_{i=1}^n{(x+k_i)},\quad\text{where } \sum_{i=1}^{n}{k_i}=0,\quad \text{and } x+k_i>0\text{ for all } i
$$</p>
<p>(and the equality only holds when all the $k_i=0$).</p>
<p>It seems intuitively obvious (when $... | Community | -1 | <p>The AM-GM inequality gives us
$$\prod_i (x+k_i)^{1/n} \leq {1\over n}\sum_i (x+k_i)=x.$$
Now take the $n$th power of both sides. </p>
|
3,867,197 | <p>Let <span class="math-container">$A$</span> be the following matrix</p>
<p><span class="math-container">$$\left(
\begin{array}{ccc}
1 & 0 & x \\
0 & 1 & y \\
x & y & 1
\end{array}
\right)$$</span></p>
<p>I have to prove that if, at least <span class="math-container">$x+y>\frac{3}{2}$</spa... | Will Jagy | 10,400 | <p>your matrix is symmetric real,</p>
<p>Use <a href="https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia#Statement_in_terms_of_eigenvalues" rel="nofollow noreferrer">Sylvester's Law of Inertia</a></p>
<p>Congruence:</p>
<p><span class="math-container">$$
\left(
\begin{array}{rrr}
1&0&0 \\
0&1&0... |
1,879,673 | <p>I have woven the below incomplete proof of the following claim:</p>
<blockquote>
<p><em>Claim</em>. If $X$ is completely regular and $Y$ is a compactification of $X$,
then there is a unique, continuous, surjective, closed map
$g:\beta\left(X\right)\to Y$ which is the identity on
$X$.</p>
</blockquote>
<p><... | Eric Wofsey | 86,856 | <p>There is one important fact you still haven't used: namely, that $Y$ is a compactification of $X$, which means not just that $X\subseteq Y$ and $Y$ is compact Hausdorff but that $X$ is dense in $Y$. As you have shown, $g$ is a closed map. In particular, taking $C=\beta X$, the image of $g$ is closed. But the imag... |
587,275 | <p>I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) = 2^{x}$, but now I'm stuck.</p>
<p>Here's my final step:
<strong>$\displaystyle{{\rm f}'\left(x\right)
= \lim_{h \to 0}{2^{x}\left(2^{h} - 1\right... | Alex | 38,873 | <p>One of definitions of logarithm is (see <a href="http://en.wikipedia.org/wiki/Logarithm#From_Napier_to_Euler" rel="nofollow">here</a>)
$$
\log x = \lim_{n \to \infty}\frac{x^{\frac{1}{n}}-1}{\frac{1}{n}}
$$
Hence denote $h=\frac{1}{n}$
$$
\lim_{h \to 0}\frac{2^{x+h}-2^x}{h}=2^x \lim_{h \to 0}\frac{2^h-1}{h}=2^x \log... |
587,275 | <p>I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) = 2^{x}$, but now I'm stuck.</p>
<p>Here's my final step:
<strong>$\displaystyle{{\rm f}'\left(x\right)
= \lim_{h \to 0}{2^{x}\left(2^{h} - 1\right... | Dan | 1,374 | <p>It helps here to use implicit differentiation.</p>
<p>$y = a^x$</p>
<p>Take the natural logarithm of both sides.</p>
<p>$\ln{y} = x \ln{a}$</p>
<p>Differentiate both sides.</p>
<p>$\frac{1}{y} dy = dx \ln{a}$</p>
<p>Multiply and divide.</p>
<p>$\frac{dy}{dx} = y \ln{a}$</p>
<p>Substitute the original definit... |
699,383 | <p>I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by</p>
<p>$$
x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)
$$</p>
<p>is decreasing. Is there an elegant way to show this?</p>
| Carser | 132,859 | <p>You want to show that $x_n$ is monotonically decreasing, or in other words that $\frac{d}{dn} x_n$ is always non-positive. The derivative is not very friendly looking, but you get
$$ \frac{d}{dn} x_n = \frac{-2 \sqrt{n} + 2(n + \sqrt{n}) log(\frac{1}{\sqrt{n}}+1)-1}{2(n+\sqrt{n})} $$
You can plug in some values fo... |
3,129,248 | <p>I am solving ordinary differential equation in <span class="math-container">$S'$</span> (dual to Schwartz space) given as:</p>
<p><span class="math-container">$y' + ay = \delta$</span>, where <span class="math-container">$\delta$</span> is a Dirac delta function.</p>
<p>The general solution of homogenous equation ... | Korvet | 647,164 | <p>"I am solving ordinary differential <strong><em>inhomogeneous</em></strong> equation in S′ (dual to Schwartz space) given as:
<span class="math-container">$$y' + ay = \delta(x)$$</span> -where <span class="math-container">$\delta(x)$</span> is a Dirac delta function.
The general solution of homogenous equation is <... |
3,029,778 | <p>I asked a similar question in <a href="https://math.stackexchange.com/questions/3029766/positive-definite-matrix-implies-the-infimum-of-eigenvalues-are-positive">here</a>, but actually what I want to ask is more difficult as described below:</p>
<p>Suppose <span class="math-container">$P(x): \mathbb{R} \to \mathbb{... | user1551 | 1,551 | <p>No. Consider e.g. <span class="math-container">$P(x)=\operatorname{diag}(x,\frac1x)$</span> over <span class="math-container">$\Omega=[1,+\infty)$</span>.</p>
<p>It is true, however, that if <span class="math-container">$\Omega$</span> is compact, <span class="math-container">$P$</span> is continuous and <span clas... |
13,882 | <p>Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on affine everything and will work out the general case at some future time.</p>
<p>The question that this got me think... | user717 | 717 | <p>If $k$ is algebraically closed and $X$ is a $k$-scheme locally of finite type, then the $k$-rational points are precisely the closed points. (See EGA 1971, Ch. I, Corollaire 6.5.3).</p>
<p>More generally: if $k$ is a field and $X$ is a $k$-scheme locally of finite type, then $X$ is a Jacobson scheme (i.e. it is qua... |
2,681,621 | <p>I'm trying to calculate the following limit:</p>
<p>$$\lim_{x\to\pi} \dfrac{1}{x-\pi}\left(\sqrt{\dfrac{4\cos²x}{2+\cos x}}-2\right)$$</p>
<p>I thought of calculating this:</p>
<p>$$\lim_{t\to0} \dfrac{1}{t}\left(\sqrt{\dfrac{4\cos²(t+\pi)}{2+\cos(t+\pi)}}-2\right)$$</p>
<p>Which is the same as:</p>
<p>$$\lim_{... | user | 505,767 | <p>From here by first order binomial expansion</p>
<p>$$\frac{1}{t}\left(\sqrt{\frac{4\cos²t}{2-\cos t}}-2\right)=\frac1t(2\cos t(1-(1-\cos t))^{-\frac12}-2)\sim\frac1t(2\cos t(1+\frac12(1-\cos t))-2)=\frac1t(2\cos t+\cos t-\cos^2t-2)=\frac{-\cos^2t+3\cos t-2}{t}=\frac{(\cos t-1)(2-\cos t)}{t^2}\cdot t\to -\frac12 \cd... |
544,008 | <p>We know since $\mathbb{Q}$ is countable that there exist a bijection $f : \mathbb{Z} \to \mathbb{Q} $. If we view $\mathbb{Q}$ and $\mathbb{Z}$ are topological subspaces of $\mathbb{R}$, are theo homeomorphic??</p>
| Don Shanil | 103,948 | <p>First pick a topology. So in this case I assume its the induced topology. Now any topological invariant will give an obstruction to a homeomorphism. For example, $\mathbb{Q}$ is everywhere dense in $\mathbb{R}$, but $\mathbb{Z}$ is not. So the answer is no, they are not homeomorphic.</p>
|
2,607,090 | <p>I have a function for which I know:</p>
<p>$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Assuming that $f$ is a polynomial, how do I find the general expression for $f$? After many minutes of fiddling I eventually found that this general expression works:</p>
<p>$f(N) = \frac{N(... | Thomas Pastor | 518,233 | <p>This is called <a href="https://en.wikipedia.org/wiki/Regression_analysis" rel="nofollow noreferrer">Regression</a>.</p>
<p>$$f(N) = f1(N)x + f2(N)y$$</p>
<p>First, you need to define what is the desired form of your expression, or what you mean "simplest". For example, people use linear form $\hat{f_1}(N) = a_1 N... |
3,290,199 | <p>If I throw a fair dice <span class="math-container">$12$</span> times, the expected number of <span class="math-container">$6$</span> is <span class="math-container">$2$</span> i.e <span class="math-container">$6$</span> is expected to appear <span class="math-container">$2$</span> times when the dice is thrown... | Mohammad Riazi-Kermani | 514,496 | <p>No, the function does not have to be bounded to have an integral.</p>
<p>Consider <span class="math-container">$$ \int _0^1 \frac {dx}{\sqrt x}$$</span> which is an improper integral because the integrand is not bounded on <span class="math-container">$(0,1)$</span>.</p>
<p>However the anti derivative is <span cl... |
3,290,199 | <p>If I throw a fair dice <span class="math-container">$12$</span> times, the expected number of <span class="math-container">$6$</span> is <span class="math-container">$2$</span> i.e <span class="math-container">$6$</span> is expected to appear <span class="math-container">$2$</span> times when the dice is thrown... | eyeballfrog | 395,748 | <p>If <span class="math-container">$f$</span> is continuous on the interval, no additional condition is needed for it to have an antiderivative. Pick any point <span class="math-container">$c$</span> in the interval and <span class="math-container">$\int_c^x f(x')dx'$</span> will be an antiderivative, since <span class... |
2,244,423 | <p>The function given is $f(x) = \sqrt[3]{{x}^2(2-x)}$.</p>
<p>Can anybody help me to find all asymptotes of this function. I know it doesn't have a vertical asymptote and I know that it's horizontal asymptote is $\sqrt[3]{-1}$, but I don't know how to find asymptote of the slope.</p>
<p>I'd prefer if someone could h... | Community | -1 | <p>You want to compute</p>
<p>$$k=\lim_{x\to\infty}\left(\sqrt[3]{{x}^2(2-x)}+x\right).$$</p>
<p>To get rid of the cubic root, you can multiply by the conjugate trinomial and get</p>
<p>$$k=\lim_{x\to\infty}\left(\frac{{x}^2(2-x)+x^3}{\sqrt[3]{{x}^2(2-x)}^2-\sqrt[3]{{x}^2(2-x)}x+x^2}\right).$$</p>
<p>The numerator ... |
3,320,830 | <p>I was wondering if the inequality
<span class="math-container">$$\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$$</span> holds for stochastic integral. In fact, I don't see such a property in any book, neither on Google, so I have some doubt. What do you think ?</p>
| Parcly Taxel | 357,390 | <p>A Raipur-bound train is only going to meet Nagpur-bound trains. And we can draw a diagram for that:</p>
<p><img src="https://i.stack.imgur.com/v129m.png" alt=""></p>
<p>The answer is <span class="math-container">$12$</span> trains.</p>
|
1,221,056 | <p>Have assigment and will use it as example, found solution computationaly, want to understand idea.</p>
<p>It is about <em>SubBytes</em> procedure in AES, particulary about finding inverse of polynomial.</p>
<p>Suppose we have element $A=x^5+1$ in finite field $F=\mathbb{Z}_2[x]/x^8+x^4+x^3+x+1$ and it is required ... | lab bhattacharjee | 33,337 | <p>$$\cos^2x=1-\sin^2x=(1-\sin x)(1+\sin x)\iff\frac{\cos x}{1+\sin x}=\frac{1-\sin x}{\cos x}$$</p>
|
1,692,346 | <p>I have heard of a statement like this:</p>
<blockquote>
<p>A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time.</p>
</blockquote>
<p>Is this possible (mathematics wise)?</p>
| Dan Christensen | 3,515 | <p>You are overthinking this. Yes, in draining a 100 litre tank full of gasoline, you can imagine that infinitely many events occur: At some point in time for example, the tank will be (1) 1/2 full, and (2) 1/4 full, and (3) 1/8 full, and so on. But we can measure the rate at which the tank is being emptied, in units o... |
1,039,474 | <p>Solve the equation $x^4 - 14x^3 + 50x^2 -14x + 1 = 0$. <br/> I am not sure about how to best proceed, and would like a solution that does not involved the generalised quartic formula.</p>
| sciona | 195,458 | <p><strong>Hint:</strong> First observe the equation is palindromic. Divide throughout with $x^2$ and rewite it as a quadratic in $\left(x+\dfrac{1}{x}\right)$.</p>
|
1,039,474 | <p>Solve the equation $x^4 - 14x^3 + 50x^2 -14x + 1 = 0$. <br/> I am not sure about how to best proceed, and would like a solution that does not involved the generalised quartic formula.</p>
| Varun Iyer | 118,690 | <p>A more detailed solution:</p>
<p>If we divide the equation by $x^2$:</p>
<p>$$\frac{x^4}{x^2} - \frac{14x^3}{x^2} + \frac{50x^2}{x^2} - \frac{14x}{x^2} + \frac{1}{x^2} = x^2 - 14x + 50 - \frac{14}{x} + \frac{1}{x^2}$$</p>
<p>Then, combining like terms, we notice that:</p>
<p>$$x^2 + \frac{1}{x^2} - 14\left(x+\fr... |
3,370,076 | <p>The total mechanical energy is conserved when a ball is dropped from a height of 4.00 <span class="math-container">$\mathit{m}$</span>, and it makes a elastic collision with the ground. Assuming no non-conservative forces are acting find the period of the ball. g of course is 9.81.</p>
<p><span class="math-containe... | Hagen von Eitzen | 39,174 | <p>Let <span class="math-container">$\beta(x)=a^{-1}x$</span>. Then <span class="math-container">$\alpha\circ \beta$</span> and <span class="math-container">$\beta\circ \alpha$</span> are both the identity map.</p>
|
301,264 | <p>Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. </p>
<p>I am a physics undergrad, and I wish to take up a course on Introduction to Algebraic Topology for the next sem, which basically teac... | Sigur | 31,682 | <p>For sure you'll need <em>continuous functions</em>, <em>homeomorphisms</em>, <em>connectedness</em>, <em>compactness</em>, <em>coverings</em> and many others.</p>
|
635,351 | <p>It is well known that if a series $\sum\limits_{k= 0}^\infty a_k$ converges, then $a_k \to 0$. </p>
<p>However, this is not true for integrals. What makes them different? Is it simply that they are "smoother?" Is there a rigorous way to explain this difference?</p>
| Glen O | 67,842 | <p>As with so many things in Mathematics, the actual identification of the right way to figure something out is actually more of an art than a science (although trial and error will often get you there).</p>
<p>A good rule of thumb with integration of functions that are products of trig is that, if you can't see an ob... |
114,733 | <p>Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?</p>
| WimC | 25,313 | <p>You can also check it explicitly:</p>
<p>$$
\left| \frac{z-1}{z+1} \right|^2 = \frac{z-1}{z+1}\cdot\frac{\overline{z}-1}{\overline{z}+1} = \frac{|z|^2-2 \Re(z) +1}{|z|^2+2 \Re(z)+1} < 1.
$$</p>
<p>The last inequality follows simply because $\Re(z) > 0$ and so the numerator is smaller than the denominator.<... |
764,947 | <p>I want to solve the following exercise:<br/>
<br/>
Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic.<br/>
$E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. <br/>
<br/>
I am trying to find a change of variables $(x,y)\mapsto(x',y')$ transforming the Weierstra... | Álvaro Lozano-Robledo | 14,699 | <p>Well, you must have the wrong equations, because they are <strong>not isomorphic</strong>. The $j$-invariant classifies elliptic curves up to isomorphism (over $\mathbb{C}$), and the $j$-invariants of these curves are $432/7$ and $-64/25$, respectively. Since they are distinct, they are not isomorphic.</p>
<p>In li... |
1,748,547 | <p>Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. </p>
<p>I know that $(a_1,b_1), ...,(a_n,b_n)$ form an open covering of $[a,b]$, and my thought is to show the inequality by mathematical induction, but not sure ho... | Eman Yalpsid | 94,959 | <p>Take any $n$ element cover.
The case $n=1$ is clear. Assume $n >1$.</p>
<p>If no two intervals intersect each other, then notice that no $a_i$ or $b_i$ is covered, therefore $a_i ,b_i \in \mathbb R \setminus [a,b]$ for all $i$. But $[a,b]$ is covered so there has to be an $i$ such that $a_i < a < b < b... |
2,235,610 | <p>I need some help for the proof of the uniformization theorem (Silverman's Advanced Topics ...).</p>
<p>If we have $G_{4}(\Lambda_{1})=G_{4}(\Lambda_{2}) $ and $ G_{6}(\Lambda_{1})=G_{6}(\Lambda_{2})$ (with $\Lambda_{1},\Lambda_{2}$ two lattices and $G_{n}$: Einsenstein serie).</p>
<p>Why we have $\Lambda_{1}=\Lam... | Joe Silverman | 317,822 | <p>It might be easiest to first note that the $j$-invariants $j(\Lambda_1)=j(\Lambda_2)$ are equal and use the theorem that the $j$-invariant defines an injective map from the space of lattice modulo homothety to the affine line. Thus the equality $j(\Lambda_1)=j(\Lambda_2)$ implies that $\Lambda_1=c\Lambda_2$ for some... |
660,259 | <p>$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$.</p>
<p>is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known
constant and $U$ has a uniform distribution on the interval $(0, 1)$, transform $U$ to obtain a random variable with the sa... | copper.hat | 27,978 | <p>Assume $b>0$.</p>
<p>Let $\phi(\alpha) = p \{ y | y \le \alpha \} = \int_{-\infty}^\alpha f(y) dy = \begin{cases} 0, & \alpha <b \\ 1-{b \over \alpha}, & \alpha \ge b\end{cases}$.
Note that the restricted $\phi:[b,\infty) \to [0,1)$ is a bijection, and we have
$\phi^{-1}:[0,1) \to [b,\infty)$ is given... |
1,507,290 | <p>Kindly help me understand this statement made by my prof.</p>
<blockquote>
<p>The identity matrix I has the property that any non zero vector <span class="math-container">$V$</span> is an eigenvector of eigenvalue <span class="math-container">$1$</span>.</p>
</blockquote>
<p>My assumption of this statement is that t... | Ben Grossmann | 81,360 | <p>From your question, it seems that you don't understand what eigenvectors are.</p>
<p>If $A$ is a matrix, then we call $v$ an eigenvector if it is not zero and $Av=\lambda v$ for some constant (that is, some scalar) $\lambda$ such that $Av=\lambda v$. The constant $\lambda$ is called an eigenvalue of $A$.</p>
<p>No... |
1,507,290 | <p>Kindly help me understand this statement made by my prof.</p>
<blockquote>
<p>The identity matrix I has the property that any non zero vector <span class="math-container">$V$</span> is an eigenvector of eigenvalue <span class="math-container">$1$</span>.</p>
</blockquote>
<p>My assumption of this statement is that t... | upe | 459,399 | <h2>Eigenvectors & Eigenvalues</h2>
<p><a href="https://www.youtube.com/watch?v=PFDu9oVAE-g" rel="nofollow noreferrer">3Blue1Brown's video on eigenvectors and eigenvalues</a> explains the eigenvectors and eigenvalues visually.</p>
<p>In general, matrix-vector multiplication <span class="math-container">$Av = b$</sp... |
550,659 | <blockquote>
<p>A space <span class="math-container">$X$</span> is locally metrizable if each point <span class="math-container">$x$</span> of <span class="math-container">$X$</span> has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space <span class="math-container">$X$</spa... | D Wiggles | 103,836 | <p>For every $x\in X$, there exists a neighborhood $U_x$ which is metrizable. These neighborhoods cover $X$, i.e., $X=\bigcup_x U_x$. Now use the definition of compactness to reduce this to a finite union, $X=U_1\cup\ldots\cup U_n$. Each of these sets is metrizable, so pick metrics which are defined locally on each $U_... |
135,936 | <p>I need this one result to do a problem correctly.</p>
<p>I want to show that for any $b \in \mathbb{C}$ and $z$ a complex variable:</p>
<p>$$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$ </p>
<p>My attempts have only led me to conclude that </p>
<p>$$ |z^2 + b^2| > \frac{|z|^{2} + |b|^{2}}{2}$$ </p>
| Tomarinator | 21,832 | <p>we know ,(from vector algebra) that</p>
<p>$$ |z^2 + b^2| \geq |z^{2}| - |b^{2}|$$</p>
<p>and, that
for any complex number $x$,</p>
<p>$$|x^{2}| \geq |x|^{2}$$ </p>
<p>therefore,
$$ |z^2 + b^2| \geq |z^{2}| - |b^{2}| \geq |z|^{2} - |b|^{2}$$</p>
<p>hence,
$$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$</p>
|
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | JTP - Apologise to Monica | 64 | <p>This is a case where you might be looking for a distinction that's pretty subtle.</p>
<p>By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the $(0,y_0)$ p... |
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | Dan Fox | 672 | <p>This questions reflects the dangers in over formalization of the language used to discuss simple things. A pedantic speaker might distinguish between the <em>y-intercept b</em> and the <em>intercept (0, b)</em> (although <em>intersection point (0, b)</em> might be a better name for the latter), but very little is ga... |
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | svavil | 6,275 | <p><a href="https://matheducators.stackexchange.com/a/14387/6275">JoeTaxpayer's answer</a> says the distinction is subtle. To me, the distinction is non-existent.</p>
<p>I don't see any benefit in discerning two concepts that are </p>
<ol>
<li>Closely related</li>
<li>Have a trivial bijection between the two concepts... |
3,465,945 | <p>Prove that <span class="math-container">$\inf f(A) \leq f( \inf A)$</span> if <span class="math-container">$f: [-\infty, + \infty] \to \mathbb{R}$</span> is continuous and <span class="math-container">$A \neq \emptyset$</span> is a subset of <span class="math-container">$\mathbb{R}$</span>.</p>
<p>Attempt;</p>
<p>... | user284331 | 284,331 | <p>Since the domain of <span class="math-container">$f$</span> is <span class="math-container">$[-\infty,\infty]$</span>, something is worth to mention.</p>
<p>For nonempty subset <span class="math-container">$A$</span> of <span class="math-container">$\mathbb{R}$</span>, <span class="math-container">$\inf A\ne\emptys... |
3,093,660 | <p>This is an introducory task from an exam. </p>
<p><strong>If</strong> <span class="math-container">$z = -2(\cos{5} - i\sin{5})$</span>, <strong>then what are:</strong></p>
<p><span class="math-container">$Re(z), Im(z), arg(z)$</span> and <span class="math-container">$ |z|$</span>?</p>
<p>First of all, how is it p... | Theo Bendit | 248,286 | <p>Currently, the number is not in polar form, as it should be in the form <span class="math-container">$r(\cos(\theta) + i \sin(\theta))$</span>, where <span class="math-container">$r \ge 0$</span>. Note the <span class="math-container">$+$</span> sign, and the non-negative number <span class="math-container">$r$</spa... |
4,150,320 | <p>I need to prove <span class="math-container">$\displaystyle \lim _{x\to 2-} \left(\frac{|x-2|}{x^2-4}\right)=\frac{-1}{4}$</span></p>
<p>I know the definition <span class="math-container">$\forall \varepsilon >0, \exists \delta >0, 0>2-x>\delta$</span> then <span class="math-container">$\left|\left(\dfra... | Paul Sinclair | 258,282 | <p>The main thing here is that they've apparently chosen to prefer using <span class="math-container">$Q$</span> to using <span class="math-container">$Q_2$</span>, so they rewrite <span class="math-container">$Q_2 = Q - Q_1$</span>, and substitute for <span class="math-container">$Q_2$</span> in your calculation (FYI ... |
672,736 | <p>Let $A = \begin{bmatrix}1&2&1\\0&1&0\\1&3&1\end{bmatrix}$. Find the eigenvalues of $A$.</p>
<p>I think I got a pretty steady ground on how I approached this, I just have some difficulty getting the right answer.</p>
<p>What I have done so far:</p>
<p>$P(\lambda) = det(A - \lambda I)$</p>
... | copper.hat | 27,978 | <p>A trial & error approach:</p>
<p>Note that $A(e_1+e_2) = 2(e_1+e_2)$, and $A(e_1-e_2) = 0$. Note also that $A A^T \neq A^T A$, hence $A$ not normal and cannot be orthogonally diagonalized (so I can just look for a vector normal to the other two). </p>
<p>Try using the basis (or rather the inverse) $P= \begin{b... |
109,569 | <p>Let's say I have a complex valued matrix $\begin{pmatrix}1+I&2+2I&3+3I\\4+4I&5+5I&6+6I\end{pmatrix}$ represented by a list:</p>
<pre><code> list = {{1 + I, 2 + 2 I, 3 + 3 I}, {4 + 4 I, 5 + 5 I, 6 + 6 I}}
</code></pre>
<p>I know how to plot each point of the matrix on the complex plane:</p>
<pre><c... | Leonid Shifrin | 81 | <p>While the concerns about performance degradation may in many cases be unwarranted, here is a way that would avoid double - traversal:</p>
<pre><code>exp /. b[a_]*c_f :> With[{res = c /. d[a] :> e}, res /; res =!= c]
</code></pre>
<p>This uses the semantics of local variables shared between the body and the c... |
4,141,378 | <p>In this equation quasi-linear eq <span class="math-container">$\Big\{\exp(f(x,y))\dfrac{\partial f(x,y)}{\partial x} + \dfrac{y}{x} \dfrac{\partial f(x,y)}{\partial y} = 1\Big\}$</span> how <span class="math-container">$f$</span> changes based on <span class="math-container">$x$</span> and <span class="math-containe... | DanielWainfleet | 254,665 | <p>A topological space is a pair <span class="math-container">$(X,T)$</span> where <span class="math-container">$T$</span> is a collection of some or all of the subsets of X such that</p>
<p>(i). <span class="math-container">$\emptyset\in T$</span> and <span class="math-container">$X\in T,$</span></p>
<p>(ii). If <span... |
1,700,689 | <p>Let $A, B$, and $C$ be sets. If $A\backslash B$ is a subset of $C$, then $A\backslash C$ is a subset of $B$. Is this a direct proof where I let $x$ be an element of $A$ and then work from there? I can't seem to figure out all of the cases. Thanks for help in advance.</p>
| Community | -1 | <p>You can do it with a direct proof style ( let x be...) but you may also prove it manipulating only sets :
We have $A\setminus B = A \cap B^c \subset C\implies C^c \subset A^c \cup B \implies C^c \cap A \subset A \cap(A^c \cup B)=(A\cap A^c)\cup (A\cap B)=\emptyset \cup (A\cap B)=A\cap B \subset B$ so we finally hav... |
272,846 | <p>Suppose I have a List of numbers:</p>
<pre><code>num = Range[5]
</code></pre>
<p>I want to combine the second and the third element into a sublist to get the result as {1,{2,3},4,5}.<br />
I tried using this:</p>
<pre><code>MapAt[List, num, {{2}, {3}}]
</code></pre>
<p>which is not giving me the desired result. What... | Daniel Huber | 46,318 | <p>Another possibility;</p>
<pre><code>num = Range[5];
num[[2 ;; 3]] = {num[[2 ;; 3]], Hold@Nothing[]};
num = num // ReleaseHold
(* {1, {2, 3}, 4, 5} *)
</code></pre>
|
272,846 | <p>Suppose I have a List of numbers:</p>
<pre><code>num = Range[5]
</code></pre>
<p>I want to combine the second and the third element into a sublist to get the result as {1,{2,3},4,5}.<br />
I tried using this:</p>
<pre><code>MapAt[List, num, {{2}, {3}}]
</code></pre>
<p>which is not giving me the desired result. What... | lericr | 84,894 | <p>For the update with SoundNote, I recommend that you simply write your own "fixing" function:</p>
<pre><code>FixSoundNote[SoundNote[pitch_, start_, end_, style_]] :=
SoundNote[pitch, {start, end}, style]
</code></pre>
<p>Usage:</p>
<pre><code>badMusic = SoundNote["CSharp", 0.1, 0.2, "Violi... |
3,014,453 | <p>If there is a number somewhere between 0 and 100 and you have to find it with the least attempts possible. Every attempt consists of you checking if the number is smaller (or bigger) than a number in the said interval (0 to 100). My guess would be you start with the half way point.</p>
<p>Is it smaller than 50?
yes... | Bram28 | 256,001 | <p>Using the <em>exact</em> halfway point is <em>not</em> the fastest method. For example, suppose the number is <span class="math-container">$98$</span>. Then you get:</p>
<p><span class="math-container">$50 \rightarrow 75 \rightarrow 87.5 \rightarrow 93.75 \rightarrow 96.875 \rightarrow 98.4375 \rightarrow 97.65625$... |
2,890,625 | <p>Suppose $f(x)$ is differentiable on $[0,1]$, and $f(0)=0$, $f(x)\ne 0,\forall x\in(0,1)$ , Prove for every $n,m\in\mathbb{N^+}$, there exists $\xi=\xi_{n,m}\in(0,1)$ such that
$$n\cdot\frac{f'(\xi)}{f(\xi)}=m\cdot\frac{f'(1-\xi)}{f(1-\xi)}$$</p>
| Theo Bendit | 248,286 | <p>Without loss of generality, we may assume $f(x) > 0$ for all $x$, since it cannot change sign, without violating the intermediate value theorem.</p>
<p>Let $h(x) = \ln(f(x)) + \frac{m}{n} \ln(f(1 - x))$, defined over $(0, 1)$. Then,
$$h'(x) = \frac{f'(x)}{f(x)} - \frac{m}{n} \frac{f(1 - x)}{f'(1 - x)}$$
so $h'(x... |
871,412 | <p>$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$</p>
<p>Can anybody tell me, how to solve this integral ?
I know that this is related to <a href="http://www.it.uom.gr/teaching/linearalgebra/NumericalRecipiesInC/c6-9.pdf" rel="nofollow">Fresnel Integral</a> if the $\cos(x)$ term is absent. </p>
| user71352 | 71,352 | <p>If $\beta>0$</p>
<p>$c+d=2\alpha-2\beta x^{2}$</p>
<p>$c-d=2x$</p>
<p>so $c=\alpha+x-\beta x^{2}=(\alpha+\frac{1}{4\beta})-(\frac{1}{2\sqrt{\beta}}-\sqrt{\beta}x)^{2}$ and $d=\alpha-x-\beta x^{2}=(\alpha+\frac{1}{4\beta})-(\frac{1}{2\sqrt{\beta}}+\sqrt{\beta}x)^{2}$</p>
<p>then using that $\sin(c)+\sin(d)=2\s... |
1,864,604 | <p>What's the difference between $f(x)=f(a-x)$ and $f(x)=f(x-a)$ ?</p>
<p>It's a pretty simple question maybe, but I'm unable to understand this one. </p>
| Martin Kochanski | 340,970 | <p>They mean two different things, and without knowing <strong>what</strong> it is that you don't understand, it's hard to know how to explain.</p>
<p>The way to understand it is to abandon algebra. Put $a=5$ and try different values of $x$: 0, 1, 2, 3, 4, 5 and so on.</p>
<p>You will find that:</p>
<ul>
<li><p>$f(x... |
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