qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,056,616 | <p><span class="math-container">$P(x) = 0$</span> is a polynomial equation having <strong>at least one</strong> integer root, where <span class="math-container">$P(x)$</span> is a polynomial of degree five and having integer coefficients. If <span class="math-container">$P(2) = 3$</span> and <span class="math-containe... | W-t-P | 181,098 | <p>If <span class="math-container">$u$</span> and <span class="math-container">$v$</span> are integer roots of <span class="math-container">$P$</span>, then <span class="math-container">$P(x)=(x-u)(x-v)Q(x)$</span>, where <span class="math-container">$Q$</span> is a polynomial with integer coefficients. From <span clas... |
1,246,250 | <p>I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the basic idea behind the proof, but I was thinking of a particular transformation, for which I don't see why it doesn't form... | MooS | 211,913 | <p>There are no integers with infinite length (in decimal system). The image of your map is contained in the rational numbers, hence can not be all of the real interval.</p>
<p>Note that the image is not even all of the rational numbers between $0$ and $1$. For instance $\frac{1}{3}$ does not appear.</p>
|
2,368,179 | <p>Answer should be in radians
Like π/4 (45°) π(90°).
I used $\tan(A+B)$ formula and got $5/7$ as the answer, but that's obviously wrong.</p>
| Vidyanshu Mishra | 363,566 | <p>Use the formula $\tan (x+y)=\frac{(\tan x+ \tan y)}{1-\tan x\tan y}$</p>
<p>And then take inverse</p>
|
2,917,896 | <p>I think my proof is wrong but I don't know how to approach the statement differently. I hope you can help me identify where I'm mistaken/incomplete.</p>
<p>Proof:
$$\text{We need to prove: } \bigcup_{n=1}^{\infty}[3 - \frac{1}{n}, 6] = [2, 6] $$</p>
<p>$$\text{Thus, } x \in \bigcup_{n=1}^{\infty}[3 - \frac{1}{n... | drhab | 75,923 | <p>You should end with something like:$$\text{From }x\in\bigcup_{n=1}^{\infty} \left[3-\frac1n,6\right]\text{ it follows that }x\in\left[3-\frac1n,6\right]\subseteq[2,6]\text{ for some positive integer }n$$</p>
<p>That integer does not have to be $1$.</p>
|
2,546,161 | <p>I came across this interesting question in an interview:</p>
<p>Given $X$ and $Y$, these two independent standard normal. We have the following probability of $P(X>0| X+Y>0) = 0.75$. One can get this easily by draw a 2d plane and find out the required area.</p>
<p>Now, if $X$ and $Y$ are joint normal with co... | Shashi | 349,501 | <p>$\newcommand{\PM}{\mathbb{P}}$I don't know if the elaboration I provide is an intuitive way of getting the answer. I think it is clear from your question that we have $X\sim N(0,1)$ and $Y\sim N(0,1)$ in the second case as well. We have (by definition):
\begin{align}
\PM(X>0 |X+Y>0) = \frac{\PM(X>0, X+Y>... |
1,691,306 | <p>Find all pairs of values $a$ and $b$ that satisfy $(a+bi)^2 = 48 + 14i$</p>
<p>Here's what I have so far:</p>
<p>$$\begin{align}
z^2 &= 48 + 14i = 50 \operatorname{cis} 0.2837\\
z &= \sqrt{50} \operatorname{cis} 0.1419 = 7 + i \\
z &= \sqrt{50} \operatorname{cis} 3.2834 = -7 - i\\
a &= ± 7 \\
b &am... | lab bhattacharjee | 33,337 | <p>$$48+14i=(a+ib)^2=a^2-b^2+i(2ab)$$</p>
<p>Equating the imaginary parts, $2ab=14\iff ab=7$</p>
<p>Equating the real parts, $a^2-b^2=48$</p>
<p>$(a^2+b^2)^2=(a^2-b^2)^2+(2ab)^2=48^2+14^2=50^2$</p>
<p>$\iff a^2+b^2=50$ as $a,b$ are real</p>
<p>and we have $a^2-b^2=48$</p>
<p>Solve for $a^2,b^2$</p>
<p>As $ab=7&g... |
638,348 | <p><img src="https://i.stack.imgur.com/U1l29.jpg" alt="enter image description here"></p>
<p>Radius of the big triangle is $2$. ABCD is a square. What is the difference between $T_{1}$ and $(M_{1}+M_{2})$.<br>
I have solved it already though I don't know if my answer is right or wrong. My solution is very long. Is t... | Community | -1 | <p>For the first case, $r=(7,2,3)+t^3(3,-1,5)$. As $t$ varies through $\mathbb R$, $t^3$ varies through $\mathbb R$, so we have a line.</p>
<p>For the second case, $r=(-1,3,1)+t^2(5,2,-1)$. As $t$ varies through $\mathbb R$, $t^2$ varies through the non-negative reals, so we have a ray.</p>
|
638,348 | <p><img src="https://i.stack.imgur.com/U1l29.jpg" alt="enter image description here"></p>
<p>Radius of the big triangle is $2$. ABCD is a square. What is the difference between $T_{1}$ and $(M_{1}+M_{2})$.<br>
I have solved it already though I don't know if my answer is right or wrong. My solution is very long. Is t... | Joel | 85,072 | <p>There are several ways to see this. Firstly you can solve each $t^3$ in terms of $x,y$ and $z$, and have a line in terms of its symmetric equation.</p>
<p>You could also re-parameterize these by making the substitution $u = t^3$, and you can see it must be a line in that case.</p>
<p>Incidentally, the second equa... |
2,762,237 | <blockquote>
<p>Does the sequence $(x_n)_{n=1}^\infty$ with $x_{n+1}=2\sqrt{x_n}$ converge?</p>
</blockquote>
<p>I'm almost positive this converges but I am not entirely sure how to go about this. The square root is really throwing me off as I haven't dealt with it at all up until now.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
518,140 | <p>What is the relation between the definition of homotopy of two functions</p>
<blockquote>
<p>"A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space $X$ with the unit i... | Brian M. Scott | 12,042 | <p>You really ought to mention some of the mathematical achievements of the $14$th century philosopher <a href="https://en.wikipedia.org/wiki/Nicole_Oresme">Nicolas Oresme</a>: he worked with fractional exponents; he was the first to prove that the harmonic series diverges; he gave in essence a formula for the sum of a... |
73,912 | <p>For my work, I am examining the values of a complex function as I vary the input according to a real parameter, and I want to both give the general plot and the plot of specific points, with labels (so one sees the direction of increase). </p>
<p>I knew from the documentation that <code>Point</code> and <code>Epil... | george2079 | 2,079 | <p>Just to show you can readily do this directly with graphics primitives:</p>
<pre><code>ourF[z_] := z^2;
parts[z_] := {Re[z], Im[z]}
ParametricPlot[parts[ourF[x + I/4]], {x, -3, 3},
Epilog -> Table[{ {PointSize[.01], Red, Point@#},
Text[Row[{"x = ", 13/10 + j/10}], #, {-2, 0}]} &@
parts[ou... |
1,454,500 | <p>I am self studying mathematics for Physics by reading book <strong>Mathematical methods in Physical Sciences</strong>. I am stuck at this problem for days:</p>
<pre><code>Prove the following by appropriate manipulations using Fact 1 to 4;
do not just evaluate the determinants.
| 1 a bc | | 1 a a^2 | ... | user236182 | 236,182 | <p>None of the two double-inequalities can hold for any $a,b,c,d\in\Bbb R^+$.</p>
<p>$$\frac{a + b}{a + b + c + d} < \frac{a}{a + c} \iff (a+b)(a+c)<a(a+b+c+d)$$</p>
<p>$$\iff \frac{a}{b}>\frac{c}{d}$$</p>
<p>The same way you get:</p>
<p>$$ \frac{a}{a+c} < \frac{b}{b + d} \iff \frac{a}{b}<\frac{c}{d}... |
3,931,672 | <p>Is there any bounded continuous map f:A to R (A is open) which can not be extended on whole R?</p>
<p>This is a question posed by myself.
My attempt: Let A=(1,2) then we can extend it. If A is finitely many intervals it can be extended. If A is countable many intervals then it can also be extended.</p>
<p>But the la... | Hagen von Eitzen | 39,174 | <p>If <span class="math-container">$A$</span> is open and neither empty nor all of <span class="math-container">$\Bbb R$</span>, then we can pick <span class="math-container">$a\in A$</span>, <span class="math-container">$b\notin A$</span>.
Wlog <span class="math-container">$a<b$</span>. Let <span class="math-contai... |
194,096 | <p>Is it possible to find an expression for:
$$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$</p>
<p>For $N=1$ we have</p>
<p>$$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = \displaystyle\sum_{k=1}^{+\infty}\frac{1}{k}$$</p>
<p>which is the (divergent) harmonic series. Thus, $S (1) = \infty$.</p... | Sasha | 11,069 | <p>Perform a partial fraction decomposition:
$$
\frac{1}{p(k)} = \frac{1}{1+k+\cdots+k^{n-1}} = \frac{1}{ \prod_{m=1}^{n-1}\left(k-\exp\left(i \frac{2 \pi}{n} m \right)\right)} = \sum_{m=1}^{n-1} \frac{1}{k-\exp\left(i \frac{2 \pi}{n} m \right)} \frac{1}{p^\prime\left(\exp\left(i \frac{2 \pi}{n} m \right)\right)}
$$
... |
4,159,060 | <p>Let <span class="math-container">$H_1, H_2$</span> be Hilbert spaces and <span class="math-container">$T:H_1\to H_2$</span>.
We say that <span class="math-container">$T$</span> is unitary if it preserves the inner product and unto.</p>
<ol>
<li>Show that the following claims are equivalent:</li>
</ol>
<p>A. <span cl... | Maximilian Janisch | 631,742 | <p>As demonstrated by the Borel–Kolmogorov paradox, it is <em>impossible</em> that the term "<span class="math-container">$\mathsf P(Y=1\mid X=x)$</span>" is defined using the event <span class="math-container">$\{X=x\}$</span> if <span class="math-container">$\mathsf P(\{X=x\})=0$</span>. Instead, the term &... |
1,555,429 | <p>Hi I am trying to solve the sum of the series of this problem:</p>
<p>$$
11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots
$$</p>
<p>I know its a geometric series, but I cannot find the pattern around this. </p>
| Brevan Ellefsen | 269,764 | <p><strong>Hint</strong></p>
<p>The geometric sequence can be rewritten as $\frac{1}{11^{-1}}, \frac{2}{11^0}, \frac{4}{11^1}, \frac{8}{11^2}...$
Notice the powers of two on top and powers of eleven on bottom</p>
|
14,515 | <p>My problem is:
What is the expression in $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$?</p>
<p>Thank you very much~</p>
| Aryabhata | 1,102 | <p>I don't think there is a "closed" form. You can give a good approximation using the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow noreferrer">Euler-McLaurin Summation</a> formula though:</p>
<p>$$\sum_{j=1}^{n} \dfrac{1}{j^2} = \dfrac{\pi^2}{6} - \dfrac{1}{n} - \dfrac{1}{2n^2... |
14,515 | <p>My problem is:
What is the expression in $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$?</p>
<p>Thank you very much~</p>
| Tyler Clark | 3,791 | <p>I am not sure if this will work or not, but maybe you could try writing the expression in terms of <a href="http://mathworld.wolfram.com/FallingFactorial.html" rel="nofollow">falling factorials</a>. Then maybe use <a href="http://mathworld.wolfram.com/SummationbyParts.html" rel="nofollow">summation by parts</a>. I a... |
956,256 | <p>If $a_n \ge 0 $ for all n, prove that $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty {a_n\over 1+a_n}$ converges. </p>
<p>Here is my attempt!</p>
<p>=> Suppose that $\epsilon \ge 0$ is given and $a_n$ converges, then for all $N\le n \le m$$$\sum_{k=n+1}^m a_k \lt \epsilon.$$
let $b_n={a_n\ove... | symmetricuser | 125,084 | <p>You made a minor mistake: $v(t)$ should be $112 - 32t$. This should make everything correct. As for the second part, ground means $s(t)=0$. So find the corresponding time and plug into $v(t)$ to find the impact velocity.</p>
|
629,347 | <p>I understand <strong>how</strong> to calculate the dot product of the vectors. But I don't actually understand <strong>what</strong> a dot product is, and <strong>why</strong> it's needed.</p>
<p>Could you answer these questions?</p>
| Cameron Williams | 22,551 | <p>Dot products are very geometric objects. They actually encode relative information about vectors, specifically they tell us "how much" one vector is in the direction of another. Particularly, the dot product can tell us if two vectors are (anti)parallel or if they are perpendicular.</p>
<p>We have the form... |
3,629,186 | <p>Assume that <span class="math-container">$x=x(t)$</span> and <span class="math-container">$y=y(t)$</span>. Find <span class="math-container">$dx/dt$</span> given the other information.</p>
<p><span class="math-container">$x^2−2xy−y^2=7$</span>; <span class="math-container">$\frac{dy}{dt} = -1$</span> when <span cla... | Allawonder | 145,126 | <p>That's not correct. Letting primes denote differentiation with respect to <span class="math-container">$t,$</span> we obtain <span class="math-container">$(x^2)'-2(xy)'-(y^2)'=7',$</span> which gives <span class="math-container">$2xx'-2(xy'+yx')-2yy'=0,$</span> which simplifies to become <span class="math-container"... |
615,067 | <p>I heard that Weil proved the Riemann hypothesis for finite fields. Where can I found the details of the proof? I found the following sketch but I was unable to fill the details: </p>
<p>Motivation: I try to understand the elementary theory of finite fields but I'm not an expert of algebraic geometry, it would be ni... | Igor Rivin | 109,865 | <p>A reasonably elementary introduction is given by Gabizon and Paskin-Cherniavsky <a href="http://www.scribd.com/doc/109789239/weilcourse-pdf" rel="nofollow">here.</a></p>
|
1,376,659 | <p>Let $5=\frac ab$
$\forall\ a,b\ \epsilon\ N$. And $(a,b)=1$ <Br>
Squaring both sides, <Br>
$25b^2=a^2$ <Br>
Thus, $25|a^2$; $25|a$ <Br>
So $a=25m$ <Br>
Substituting, $25b^2=25^2m^2$ <Br>
So $b^2=25m^2$ <Br>
So $25|b$ (By the same logic used before). <Br>
But are assumption is proved to be wrong, because $25$ comes t... | mvw | 86,776 | <p>The volume of the big box is $V_B = 7\cdot 9 \cdot 11 = 693$, the total volume of the small boxes is $V_b = 77 \cdot 3 \cdot 3 \cdot 1 = 693$.</p>
<p>This means the volume of the small boxes is sufficient and we need to use all small boxes.</p>
<p>Let us try to model this problem Tetris style: </p>
<ul>
<li>We ha... |
2,936,269 | <p>How do you simplify: <span class="math-container">$$\sqrt{9-6\sqrt{2}}$$</span></p>
<p>A classmate of mine changed it to <span class="math-container">$$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$</span> but I'm not sure how that helps or why it helps.</p>
<p>This questions probably too easy to be on the Math Stack Exc... | Siong Thye Goh | 306,553 | <p><span class="math-container">\begin{align}
9 - 6\sqrt2 &= 3 (3-2\sqrt2) \\
&= 3((\sqrt2)^2 - 2(1)\sqrt{2} +1^2) \\
&= 3(\sqrt2-1)^2
\end{align}</span>
Hence,
<span class="math-container">$$\sqrt{9-6\sqrt2} = \sqrt{3}(\sqrt2 - 1)$$</span></p>
|
3,628,159 | <p>I have <span class="math-container">$1,2,\ldots, n$</span> numbers and I want pick <span class="math-container">$k$</span> of them with replacement and such that order matters. </p>
<p>So for <span class="math-container">$n=10$</span> and <span class="math-container">$k=4$</span> I can get: <span class="math-contai... | DonAntonio | 31,254 | <p>You could argue as follows: suppose <span class="math-container">$\;3\mid(4^n+5)\;$</span> isn't true for all natural numbers, and let <span class="math-container">$\;K:=\left\{\,k\in\Bbb N\;|\; 3\nmid(4^k+5)\,\right\}\;$</span> . As <span class="math-container">$\;K\neq\emptyset\;$</span> by assumption, the WOP tel... |
3,531,971 | <p>Let <span class="math-container">$T$</span> a linear operator. <span class="math-container">$T$</span> is bounded then ker(<span class="math-container">$T$</span>) is closed.</p>
<p><b>My attempt:</b></p>
<p>Let <span class="math-container">$\{x_n\}\subset \ker(T)$</span>.</p>
<p>As <span class="math-container">$... | Leonardo | 557,543 | <p>Continuing with your proof: suppose <span class="math-container">$x_n \in \text{ker}T$</span> converges to <span class="math-container">$x$</span>. You have to show that <span class="math-container">$x \in \text{ker}T$</span>. Since <span class="math-container">$T$</span> is linear and bounded then it is also conti... |
484,313 | <p>I am taking linear algebra and none of this stuff is expained. I found this helpful link <a href="http://www.math.ucla.edu/~pskoufra/M115A-Notation.pdf" rel="nofollow">http://www.math.ucla.edu/~pskoufra/M115A-Notation.pdf</a></p>
<p>but it is missing a lot of what I need to know. Just right now though what does v a... | Emily | 31,475 | <p>Sometimes, $\wedge$ is used to mean "and", whereas $\vee$ is used to mean "or." The LaTeX code for these are <code>\wedge</code> and <code>\vee</code>, respectively.</p>
<p>$\forall$ means "for all". So if you wrote $\forall x \in S$, this means literally "for every element $x$ in the set $S$."</p>
<p>$\exists$ me... |
1,397,776 | <p>Suppose $X_1,\ldots,X_n$ are iid r.v.'s, each with pdf $f_{\theta}(x)=\frac{1}{\theta}I\{\theta<x<2\theta\}$. I find the minimal sufficient statistics $(X_{(1)},X_{(n)})$. I am trying to prove it is complete. Can someone give me hint? Also are there any complete sufficient statistics in this model?</p>
| Saty | 213,298 | <p>Find $E(X_{(1)})$ and $E(X_{(n)})$. Play with them to make it $0$ i.e. find $a,b$ such that $E[aX_{(1)}+bX_{(n)}]=0$. Call $g(T)=aX_{(1)}+bX_{(n)}$. Then we have $E[g(T)]=0$ but does that mean $g(T)=0$ a.e.? No right?</p>
|
52,874 | <p>Consider a coprime pair of integers $a, b.$ As we all know ("Bezout's theorem") there is a pair of integers $c, d$ such that $ac + bd=1.$ Consider the smallest (in the sense of Euclidean norm) such pair $c_0, d_0$, and consider the ratio $\frac{\|(c_0, d_0)\|}{\|(a, b)\|}.$ The question is: what is the statistics of... | Bill Thurston | 9,062 | <p>Here's a more geometric formulation of your question:</p>
<p>On the torus $\mathbb R^2/\mathbb Z^2$, consider a long simple closed geodesic
$\overline {(0,0)(a,b)}$. It cuts the torus into a thin cylinder; the cylinder is joined
to itself by a twist by some angle to form the torus. What is the distribution of the
... |
52,874 | <p>Consider a coprime pair of integers $a, b.$ As we all know ("Bezout's theorem") there is a pair of integers $c, d$ such that $ac + bd=1.$ Consider the smallest (in the sense of Euclidean norm) such pair $c_0, d_0$, and consider the ratio $\frac{\|(c_0, d_0)\|}{\|(a, b)\|}.$ The question is: what is the statistics of... | David Feldman | 10,909 | <p>Roughly:</p>
<p>Suppose you have a fraction $a/b$ and you expand it as a simple continued fraction
$1/c_1+1/c_2\cdots+1/c_{n-1}+1/c_n$. Now truncate the last convergent and collapse
$1/c_1+1/c_2\cdots+1/c_{n-1}$ to get, say, $a'/b'$. Now consider the simple continued
fraction expansion of $b'/b$. As I recall, th... |
1,567,152 | <blockquote>
<p>Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete.</p>
</blockquote>
<p>My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ is open. Now for any $y\in X \setminus \{x\}$ and $x$ using Hausdorff property, ... | Léreau | 351,999 | <p>If you showed that, for Hausdorff spaces , all one point sets $\{x\}$ are closed, it also follows that all finite subsets $F = \{x_1,\dots, x_n \}$ are closed, since
$$
F = \bigcup_{j=1}^n \{x_j\}
$$
will be closed as a finite union of closed sets.</p>
<p>Let's apply this to the situation where $X$ is finite and H... |
3,235,300 | <p>I tried with , whenever <span class="math-container">$x > y$</span> implies <span class="math-container">$p(x) - p(y) =( 5/13)^x (1-(13/5)^{(x-y)}) + (12/13)^x (1- (13/12)^{(x-y)}) > 0 $</span>.
But here I don't understand why the answer is no.</p>
| Fred | 380,717 | <p>We have <span class="math-container">$p(x)=a^x+b^x -1$</span> with <span class="math-container">$0< a,b<1.$</span></p>
<p>Then <span class="math-container">$p'(x)= a^x \ln a +b^x \ln b$</span>. Since <span class="math-container">$ \ln a , \ln b<0$</span>, <span class="math-container">$p'(x)<0.$</span></... |
4,008,987 | <p>I am reading a math book and in it, it says, "Let <span class="math-container">$V$</span> be the set of all functions <span class="math-container">$f: \mathbb{Z^n_2} \rightarrow \mathbb{R}.$</span> I know that <span class="math-container">$\mathbb{Z^n_2}$</span> is just the cyclic group of order <span class="ma... | Arturo Magidin | 742 | <p>The fact that <span class="math-container">$\mathbb{Z}_2^n$</span> is a group is completely irrelevant here. It is being used simply as a set. So let us discuss this construction without any reference to <span class="math-container">$\mathbb{Z}_2^n$</span>.</p>
<p>Let <span class="math-container">$X$</span> be your ... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Tom Copeland | 12,178 | <p>One of the most clever and amusing, yet deep, presentations I've seen on the interdisciplinary import of the golden ratio and Fibonacci sequences is the video sequence by V. Hart "<a href="https://www.youtube.com/watch?v=ahXIMUkSXX0" rel="nofollow noreferrer">Doodling in Math: Spirals, Fibonacci, and Being a Pl... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Pietro Majer | 6,101 | <p>The golden ratio is also the order of convergence of the <a href="https://en.wikipedia.org/wiki/Secant_method" rel="nofollow noreferrer">secant method</a> .</p>
<p><em><strong>edit Dec 2022.</strong></em> It seems it has not yet been recalled the relevance of the golden ratio and Fibonacci numbers in the theory of <... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | KConrad | 3,272 | <p>In every real quadratic field <span class="math-container">$K$</span>, the unit group of its ring of integers <span class="math-container">$\mathcal O_K$</span> is known to have the form <span class="math-container">$\pm u^\mathbf Z$</span> for a unique number <span class="math-container">$u > 1$</span>, which is... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Brian Hopkins | 14,807 | <p>Clearly there are many possible answers; MathSciNet has 50 entries with "golden mean" in the title and 447 other entries with the phrase appearing "anywhere." Let me mention three in particular.</p>
<p>One of the commonly claimed applications of the Fibonacci numbers in nature is sunflower seeds... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Tony Huynh | 2,233 | <p>The golden mean also crops up in matroid theory. A matroid <span class="math-container">$M$</span> is a <em>golden mean</em> matroid if it can be represented by a real matrix such that every non-zero subdeterminant is <span class="math-container">$\pm \phi^i$</span>, for some <span class="math-container">$i \in \ma... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | Timothy Chow | 3,106 | <p>Here is an example which on the surface has nothing at all to do with Fibonacci numbers or continued fractions. Theorem 1.1 of Itai Dinur's 2021 SODA paper <em><a href="https://doi.org/10.1137/1.9781611976465.151" rel="noreferrer">Improved algorithms for solving polynomial systems over GF(2) by multiple parity-coun... |
3,667,798 | <p>Find values of <span class="math-container">$a$</span> for which the integral <span class="math-container">$$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$</span> converges</p>
<p>What i try</p>
<p><span class="math-container">$$\int^{\infty}_{0}e^{-at}\sin(7t)dt$$</span></p>
<p><span class="math-container">$$=\frac{1}{a^2... | Allawonder | 145,126 | <p>That converges for all <span class="math-container">$a>0$</span> since then we have that <span class="math-container">$$e^{-at}|\sin 7t|\le e^{-at}.$$</span></p>
<p>You can directly see that it does not when <span class="math-container">$a\le 0.$</span></p>
|
271,915 | <p>Not clear from <code>DayMatchQ</code> <a href="https://reference.wolfram.com/language/ref/MatchQ.html" rel="nofollow noreferrer">doc page</a> but doesn't seem to work for say alternatives the way <code>MatchQ</code> does, eg</p>
<p><code>DayRange[Today,DayPlus[Today,30]] // Select[DayMatchQ[#,Monday | Wednesday | Fr... | lericr | 84,894 | <p>I assume that DayMatchQ is more complicated than just matching a form. One workaround would be to use DayName:</p>
<pre><code>Select[
DayRange[Today, DayPlus[Today, 30]],
MemberQ[{Monday, Wednesday, Friday}, DayName[#]] &]
</code></pre>
|
390,640 | <p>Please help me to find a closed form for the following integral:
$$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$</p>
<p>I was told it could be calculated in a closed form.</p>
| math110 | 58,742 | <p>Find the integral
$$I=\int_{0}^{1}\ln{\left(\ln{\left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2}-1}\right)}\right)}dx$$
solution:since let $$x=e^{-y}$$
then
$$I=\int_{0}^{\infty}e^{-y}\ln{\ln{\left(e^y+\sqrt{e^{2y}-1}\right)}}dy$$
so
\begin{align*}
&\int_{0}^{\infty}e^{-x}\ln{(\ln{(e^x+\sqrt{e^{2x}-1})})}dx=_{y=\ln{(e^x... |
390,640 | <p>Please help me to find a closed form for the following integral:
$$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$</p>
<p>I was told it could be calculated in a closed form.</p>
| Nanayajitzuki | 611,558 | <p>Following the comments above, there is another path with digamma function. Recalling this identity of Euler-Mascheroni constant</p>
<p><span class="math-container">$$-\int_{0}^{\infty} {e^{-u}\ln u \>\mathrm{d}u} = \gamma$$</span></p>
<p>and</p>
<p><span class="math-container">$$\frac{\mathrm{d}(e^{-u}\tanh u)}{\... |
4,563,707 | <p>Sequence given : 6, 66, 666, 6666. Find <span class="math-container">$S_n$</span> in terms of n</p>
<p>The common ratio of a geometric progression can be solved is <span class="math-container">$\frac{T_n}{T_{n-1}} = r$</span>, where r is the common ratio and n is the</p>
<p>When plugging in 66 as <span class="math-c... | Henry | 6,460 | <p>There are many ways, including saying <span class="math-container">$$T_n=\frac23(10^n-1)$$</span> and so <span class="math-container">$$S_n=\sum\limits_{k=1}^n T_k=\frac23\left(\sum\limits_{k=1}^n 10^k - \sum\limits_{k=1}^n 1 \right)$$</span> where the sums are simple geometric series.</p>
<p>Personally I prefer</p>... |
1,903,263 | <p>I am developing an algorithm that approximates a curve using a series of linear* segments. The plot below is an example, with the blue being the original curve, and the red and yellow being an example of a two segment approximation. The x-axis is time and the y-axis is attenuation in dB. The ultimate goal is to use ... | Francesco Alem. | 175,276 | <p>An approach would be this one:</p>
<p>1) Start with only one segment that connects start and end point of the curve.</p>
<p>2) find the point on the curve where the approximation error is maximum.</p>
<p>3) if the error is below the tolerance: go to step 5; else: continue</p>
<p>4) add one point exacly where the... |
7,080 | <p>What is the right definition of the symmetric algebra over a graded vector space V over a field k?</p>
<p>More generally: What is the right definition of the symmetric algebra over an object in a symmetric monoidal category (which is suitably (co-)complete)?</p>
<p>Two possible definitions come to my mind:</p>
<p... | S. Carnahan | 121 | <p>Symmetric algebras (aka free commutative associative unital algebras) are given by a functor, and they satisfy a universal property: If M is a module over a commutative ring k and R is a commutative k-algebra, then k-algebra homomorphisms from Sym<sub>k</sub>(M) to R are in bijection with k-module maps from M to R. ... |
3,506,982 | <p>Let <span class="math-container">${X_n, n\in N}$</span> be an iid sequence of psitive rrvs and let <span class="math-container">$K$</span> be a rrv independent of this sequence and taking its values in <span class="math-container">$N$</span> with <span class="math-container">$P(K=k)=p_k$</span>. Consider the rrv <sp... | Sri-Amirthan Theivendran | 302,692 | <p>Note that
<span class="math-container">$$
Z=\sum_{n=1}^\infty X_nI(K\geq n).
$$</span>
Since the <span class="math-container">$X_n$</span> are non-negative the monotone convergence theorem together with independence of <span class="math-container">$X_n$</span> and <span class="math-container">$K$</span> imply that
<... |
2,525,498 | <p>I'm an undergrad, and I've been presented with the following problem:</p>
<blockquote>
<p>Fundamental Theorem of Arithmetic: Let $\mathbb{N}_{>0}$ be the monoid of positive
integers with binary operation given by ordinary multiplication, let $P$ be
the set of primes in $\mathbb{N}$, let $M$ be a commutativ... | Rob Arthan | 23,171 | <p>Let $f : (0, 1) \to [0, 1]$ is a continuous bijection and let's write $f[X]$ for the image of $X \subseteq (0, 1)$ under $f$. As $f$ is a bijection, there is a unique $x \in (0, 1)$ such that $f(x) = 0$. And then $f[(0, x]]$ and $f[[x, 1)]$ are connected subsets of $[0, 1]$ each having at least two elements and each... |
2,525,498 | <p>I'm an undergrad, and I've been presented with the following problem:</p>
<blockquote>
<p>Fundamental Theorem of Arithmetic: Let $\mathbb{N}_{>0}$ be the monoid of positive
integers with binary operation given by ordinary multiplication, let $P$ be
the set of primes in $\mathbb{N}$, let $M$ be a commutativ... | DanielWainfleet | 254,665 | <p>Suppose $f:(0,1)\to [0,1]$ is a continuous surjection . For $n\in \Bbb N$ the subspace $S(n)=[2^{-n},1-2^{-n}]$ is connected so its image $f(S(n))$ is connected so $f(S(n))$ is an interval. </p>
<p>For some $n_1$ there exist $x,y\in S(n_1)$ with $f(x)=0$ and $f(y)=1,$ implying that $0$ and $1$ belong to the ... |
2,877,916 | <p>Can you some one please tell how to prove Holder Space is Normed Linear Space</p>
<p>The Holder Space $C^{k,\gamma}(\bar{U})$ consisting of the all $u \in C^k(\bar{U})$ for which the norm</p>
<p>$$\|u\|_{C^{k,\gamma}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{... | Andres Mejia | 297,998 | <p>Take the square with one point removed. You can certainly radiallyretract to the boundary to just get $S^1$.</p>
<p>For $2$ points removed, put one at $(1/3,1/2)$ and $(2/3,1/2)$ (which I'll include just for concreteness.</p>
<p>Using the vertical line at $x=1/2$, you can radially retract each "half of the square... |
1,037,736 | <p>$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$</p>
<p>please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums'</p>
<p>proof:</p>
<p>if we look at $\sum \limits_{v=1}^3 v=1+2+3,\sum \limits_{v=1}^4 v=1+2+3+4,\sum \limits_{v=1}^5 v=... | Community | -1 | <p>$\displaystyle\sum_{v=1}^nv=\dfrac{1}{2}\displaystyle\sum_{v=1}^n2v=\dfrac{1}{2}\displaystyle\sum_{v=1}^n\left((v+1)^2-v^2-1\right)=\dfrac{1}{2}\left((n+1)^2-(n+1)\right)-=\dfrac{1}{2}n(n+1)$</p>
|
3,369,188 | <p>For which frequency k>f is </p>
<p><span class="math-container">$\cos (\omega fx_1)= \cos (\omega kx_1)$</span></p>
<p>only at point <span class="math-container">$x_1$</span>.</p>
<p>Like k = 9</p>
<p><span class="math-container">$\cos (\omega*1*0.1)= \cos (\omega * 9 * 0.1) \approx 0.809$</span></p>
<p>I need ... | sabeelmsk | 578,078 | <p>Hint: Here, <span class="math-container">$3x^2+5x=x(3x+5)$</span> both factors are irreducible and relatively prime.
Use Chinese reminder theorem for rings to find an isomorphism and to SO get the order of ring.</p>
|
3,369,188 | <p>For which frequency k>f is </p>
<p><span class="math-container">$\cos (\omega fx_1)= \cos (\omega kx_1)$</span></p>
<p>only at point <span class="math-container">$x_1$</span>.</p>
<p>Like k = 9</p>
<p><span class="math-container">$\cos (\omega*1*0.1)= \cos (\omega * 9 * 0.1) \approx 0.809$</span></p>
<p>I need ... | lhf | 589 | <p>Let <span class="math-container">$f=3x^{2}+5x$</span> and <span class="math-container">$I=\langle f \rangle$</span>.</p>
<ul>
<li><p><span class="math-container">$I \ni 3f = 9x^2+15x = 9x^2$</span>.</p></li>
<li><p><span class="math-container">$I \ni 5f = 15x^2+25x = 25x = 10x = -5x$</span>.</p></li>
<li><p><span c... |
164,060 | <p>When I plot the data I have using <code>ListStepPlot</code> and <code>ListLinePlot[data,InterpolationOrder -> 0]</code> I am getting two different plot. I guess there is a bug in <code>ListStepPlot</code>. </p>
<pre><code>data={{{0, 1}, {0.0582215, 2}, {0.597255, 3}, {1.17158, 4}}, {{1.17158,
4}, {1.36478, 5... | Alan | 19,530 | <p><code>ListStepPlot</code> always produces a step shape. You can decide where you want the resulting "extra bit" of the curve with a second argument.</p>
<pre><code>data = {{{1, 2}, {2, 3}}, {{3, 4}, {4, 5}}};
left = ListStepPlot[data, Left, Mesh -> Full, Frame -> True,
PlotTheme -> "Detailed", ImageSiz... |
4,304,724 | <p><strong>Here's the question</strong>: <em>Suppose there's a bag filled with balls numbered one through fifty. You reach in and grab three at random, put them to the side, and then replace the ones you took so that the bag is once again filled with fifty distinctly numbered balls. Do this five times, so you have 5... | Rohit Pandey | 155,881 | <p>First, you consider the probability of the number <span class="math-container">$1$</span> appearing three or more times. You can then multiply this probability by <span class="math-container">$50$</span>. In any 3-ball draw, the probability of <span class="math-container">$1$</span> appearing is <span class="math-co... |
2,455,428 | <p>While going through some exercises in my analysis textbook, I came up with an equation which looks like an identity. I strongly believe that this is the case, but I couldn't prove this.</p>
<blockquote>
<p>$$\sum_{0\leq k\leq n}(-1)^k\frac{p}{k+p}\binom{n}{k} = \binom{n+p}{p}^{-1}$$</p>
</blockquote>
<p>Can some... | Ivan Neretin | 269,518 | <p>The general strategy is to introduce a variable $x$ and then to deal with some polynomials and/or other power series, of which your expression is a value at $x=1$.</p>
<p>In you case it goes along these lines:
$$\sum_{0\leq k\leq n}(-1)^k\frac{p}{k+p}\binom{n}{k}=F(1),\text{ where} \\
F(x)=\sum_{0\leq k\leq n}(-1)^... |
2,332,419 | <p>What's the angle between the two pointers of the clock when time is 15:15? The answer I heard was 7.5 and i really cannot understand it. Can someone help? Is it true, and why?</p>
| B. Goddard | 362,009 | <p>At 15:15, the minute hand is 90 degrees from 12. The hour hand is $3+1/4= 13/4$ hours past 12, so the angle is $13/4$ out of $12$ hours. So it's</p>
<p>$$\frac{13/4}{12} 360 = \frac{195}{2}.$$</p>
<p>The difference between the two angles is </p>
<p>$$\frac{195}{2} - 90 = \frac{15}{2}.$$</p>
|
384,450 | <p>I don't know what this double-arrow $\twoheadrightarrow$ means!</p>
| amWhy | 9,003 | <p>From Wikipedia: <a href="http://en.wikipedia.org/wiki/Surjective_function">Surjective funtion</a></p>
<blockquote>
<p>A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domain $X$ and codomain $Y$ is surjective if for every $y$ in $Y$ there exists at least on... |
2,564,321 | <p>I'm trying to prove </p>
<p>$$e^x\leq e^a\frac{b-x}{b-a}+e^b\frac{x-a}{b-a}$$</p>
<p>for any $x\in[a,b]$. Since this looks reminiscent of the mean value theorem or linear approximations I jotted down some equations relating to those, but didn't see any way of making progress with them. I know that $e^x$ is an in... | user284331 | 284,331 | <p>Since $\exp:x\rightarrow e^{x}$ is convex, we have
\begin{align*}
\exp\left(\dfrac{b-x}{b-a}a+\dfrac{x-a}{b-a}b\right)\leq\dfrac{b-x}{b-a}\exp(a)+\dfrac{x-a}{b-a}\exp(b).
\end{align*}</p>
|
4,141,477 | <blockquote>
<p>Find <span class="math-container">$$\lim_{x\rightarrow0} x\tan\frac1x$$</span></p>
</blockquote>
<p>Now I tried to find the form of the limit (<span class="math-container">$0/0$</span> or <span class="math-container">$0\cdot \infty$</span> or <span class="math-container">$\infty/\infty$</span>), but as ... | Thomas Andrews | 7,933 | <p>Let <span class="math-container">$f(x)=x\tan(1/x)$</span> when <span class="math-container">$x\neq 0.$</span></p>
<p>If <span class="math-container">$a_n=\frac{1}{2\pi n}$</span> then <span class="math-container">$f(a_n)=0$</span> for all <span class="math-container">$n.$</span> Thus <span class="math-container">$$\... |
1,252,167 | <p>I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. </p>
<p>It seems that functions can be manipulated as vectors as long as they are not interpreted as having real values.<br>
Suppose the solution space of a linear homogeneous di... | Ben Blum-Smith | 13,120 | <p>As often happens when I write questions here, I realized something while writing. Here I actually realized the answer.</p>
<p>It is <strong>no</strong>.</p>
<p>Here is an open cover of $\mathbb{R}$ with no irredundant subcover:</p>
<p>$\Lambda=\mathbb{N}$.</p>
<p>$U_n = (-n,n)$.</p>
<p>Because $U_1\subset U_2\s... |
1,824,966 | <p>Ok, I was asked this strange question that I can't seem to grasp the concept of..</p>
<blockquote>
<p>Let $T$ be a linear transformation such that:
$$T \langle1,-1\rangle = \langle 0,3\rangle \\
T \langle2, 3\rangle = \langle 5,1\rangle $$
Find $T$.</p>
</blockquote>
<p>Is there suppose to be a funct... | jugglingmike | 341,620 | <p>Assuming $T:\mathbb{K}^2 \rightarrow \mathbb{K}^2$ some vector field $\mathbb{K}$, the required transformation can be expressed as a matrix. To do this recall $T<1,0>$ gives the first column and $T<0,1>$ gives the second. <br/>
Then $T<1,0> = \frac{1}{5}(3<0,3>+<5,1>)=<3,2>$ and... |
37,052 | <p>This is my first question with mathOverflow so I hope my etiquette is up to par here.</p>
<p>My question is regarding a <span class="math-container">$3\times3$</span> magic square constructed using the la Loubère method (see <a href="http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of... | BigBill | 5,210 | <p>The answer is provided by the article</p>
<ul>
<li>Edward G. Effros and Mihai Popa, <em>Feynman diagrams and Wick products associated with q-Fock space</em>, PNAS 100 (15) (2003) 8629-8633, <a href="https://doi.org/10.1073/pnas.1531460100" rel="nofollow noreferrer">https://doi.org/10.1073/pnas.1531460100</a></li>
</... |
2,445,023 | <p>I have a trouble in calculating which function grows faster. </p>
<p>$f(n) = 3\log_4 n + \sqrt{n} + 3 \\
g(n) = 4\log_3 n + \log n + 200$</p>
<p>Can someone let me know how to solve this? </p>
| videlity | 70,729 | <p>$\log n$ grows slower than any $n^d$, in particular $\sqrt{n}$.
Then, look at the terms which grow the fastest in $f$ and $g$.
It is clear that $f$ will grow faster because it has the $\sqrt{n}$ term.</p>
<p>You can show this concretely by considering
$$\lim_{n\to\infty} \frac{f(n)}{g(n)}$$
and showing it goes to ... |
3,663,526 | <p><span class="math-container">$F(x)=\begin{cases} x^3+5, & x\ge 1\\ x^3+2, & 0\leq x<1\\ x^3, & x<0 \end{cases}$</span></p>
<p>Let <span class="math-container">$\mu_{F}$</span> be the Lebesgue-Stieltjes measure associated with <span class="math-container">$F$</span>. Find the lebesgue decomposition... | Saptak Bhattacharya | 734,601 | <p>This function is not absolutely continuous,to be precise.For one thing, absolutely continuous functions need to be continuous,which is clearly not the case here.Secondly,you can think of Lebesgue decomposion theorem in terms of projections on cones.You call two measures mutually orthogonal if either of them is zero ... |
3,663,526 | <p><span class="math-container">$F(x)=\begin{cases} x^3+5, & x\ge 1\\ x^3+2, & 0\leq x<1\\ x^3, & x<0 \end{cases}$</span></p>
<p>Let <span class="math-container">$\mu_{F}$</span> be the Lebesgue-Stieltjes measure associated with <span class="math-container">$F$</span>. Find the lebesgue decomposition... | Ian | 83,396 | <p>You first try to subtract off <span class="math-container">$3x^2 dm(x)$</span> from <span class="math-container">$\mu_F$</span>, since that's definitely an absolutely continuous measure that's in there.</p>
<p>What you're left with is now <span class="math-container">$\mu_G$</span> where <span class="math-container... |
49,281 | <p>I have two ingredients here:</p>
<ul>
<li>a big dataset contained in a list, with ~ 20M values. </li>
<li>a function that takes as each element of the list as input and yields True or False</li>
</ul>
<p>I want to save somewhere the elements of the list that yielded True.
Usually I would do something like that:</p... | ciao | 11,467 | <p>Rather than partition the whole list which will just gobble space (Edit: Turns out that's false in general for simple partitioning, but nonetheless the way you're doing it in your example can cause data to be missed unless the number of partitions is an exact divisor of the length of the list - you'd need <code>Part... |
261,410 | <p>Let $z_1,z_2,\dots,z_n\in\Bbb{C}$ be distinct and $w_1,w_2,\dots,w_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z_j)=w_j,\qquad g(z_j)=\bar{w}_j \qquad\text{for $1\leq j\leq n$}.$$
Define $\Omega(z)=\prod_{j=1}^n(z-z_j)$. The following puzzles me.</p>
<blockquo... | Markus Sprecher | 100,908 | <p>This Matlab program shows that your conjecture is in general not true for $n\geq 3$</p>
<pre><code>n = 3;
z = rand(1, n) + 1i * rand(1, n)
w = rand(1, n) + 1i * rand(1, n)
f = polyfit(z, w, n-1);
g = polyfit(z, conj(w), n - 1);
% note that Omega(z_i) = 0 and Omega(z) = z^n+q(z) where deg(q)<=n-1 and
% hence q... |
2,618,746 | <p>The distance between two stations $X$ and $Y$ is 220 km.</p>
<p>Trains $P$ and $Q$ leave station $X$ at 7 am and 8:15 am respectively at the speed of 25 km/hr and 20 km/hr respectively for journey towards $Y$.</p>
<p>Train $R$ leaves station $Y$ at 11:30 am at a speed of 30 km/hr for journey towards $X$. </p>
<p>... | Michael Hoppe | 93,935 | <p>Hint: start by analyzing the situation at 11:30</p>
|
2,604,178 | <p>Let $(G=(a_1,...,a_n),*)$ be a finite Group. Define for a element $a_i \in G$ a permutation $\phi = \phi(a_i)$ by left multiplication:</p>
<p>$$
\begin{bmatrix}
a_1 & a_2 & ... & a_n \\
a_i*a_1 & a_i*a_2 & ... & a_i*a_n \\
\end{bmatrix}
$$
I am struggling to understand why this is the permu... | user3794724 | 521,240 | <p>In some cases, the Newton's Method does not work. For example, if you encounter a stationary point in the process (division by zero). </p>
<p>Your example is another one in which Newton's Method does not work, because starting with a real number you only go through real numbers in the process. BUT you could start w... |
3,149,110 | <p>I am learning algebraic number theory, the exercises are so hard for me, could you please recommend me a book with answers? Many thanks!</p>
| vxnture | 652,366 | <p>Murty & Esmonde's <em>Problems in Algebraic Number Theory</em> (available <a href="http://148.206.53.84/tesiuami/S_pdfs/Problems%20in%20Algebraic%20Number%20Theory.pdf" rel="nofollow noreferrer">here</a> as a pdf) is an excellent source of problems with solutions. However, as someone pointed out in the comments,... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Phil Tosteson | 52,918 | <p>There is <a href="https://en.wikipedia.org/wiki/Euler_characteristic" rel="noreferrer">Euler's formula</a> <span class="math-container">$$V - E + F = 2.$$</span> Today, we might not think of the hypotheses as being especially tricky. But Lakatos's classic <a href="https://www.jstor.org/stable/685347" rel="noreferre... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Noah Schweber | 8,133 | <p>This is one which I've seen trip up a number of students when first learning the material: the hypothesis of <strong>admissibility</strong> <em>(or <strong>acceptability</strong> - I learned the latter, but the former seems more common)</em> in the context of numberings of unary partial computable functions (or equi... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Asaf Karagila | 7,206 | <blockquote>
<p><strong>Theorem.</strong> Assuming the axiom of choice, the countable union of countable sets is countable.</p>
</blockquote>
<p><em>Proof.</em> Let <span class="math-container">$\{A_n\mid n\in\Bbb N\}$</span> be a family of countable sets, and so we can write <span class="math-container">$A_n$</span... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Oscar Lanzi | 86,625 | <p>How about the classic equality <span class="math-container">$0.999...=1$</span>?</p>
<p>Proofs of this fact either assume the series defined by this infinite decimal representation converges, or initially prove convergence using the fact that the common ratio of the geometric series has norm less than unity. But the... |
3,623,924 | <p>Trying to solve the following problem:</p>
<p>Let <span class="math-container">$f(x)$</span> be a continuous real-valued function on <span class="math-container">$[0,3]$</span>. Given any <span class="math-container">$\varepsilon>0$</span> prove there exists a polynomial, <span class="math-container">$p(x)$</spa... | Sebathon | 482,453 | <p>Your problem is that <span class="math-container">$p=\lim P_{n}(x)$</span> may not be a polynomial. I understand your idea and i fixed: By Weierstrass's theorem, there exists a sequence of polynomials such that <span class="math-container">$P_{n} \to f$</span> uniformly. So, given <span class="math-container">$\vare... |
1,041,177 | <p>Proof that if $p$ is a prime odd and $k$ is a integer such that $1≤k≤p-1$ , then the binomial coefficient</p>
<p>$$\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod p$$</p>
<p>This exercise was on a test and I could not do!!</p>
| André Nicolas | 6,312 | <p>Let $a=\binom{p-1}{k}$. Then
$$a k!=(p-1)(p-2)(p-3)\cdots (p-k).$$
The $i$-th term on the right-hand side is congruent to $-i$ modulo $p$.
Thus
$$ak!\equiv (-1)^k k!\pmod{p}.$$
Now since $k!$ is not divisible by $p$ we can cancel.</p>
|
1,041,177 | <p>Proof that if $p$ is a prime odd and $k$ is a integer such that $1≤k≤p-1$ , then the binomial coefficient</p>
<p>$$\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod p$$</p>
<p>This exercise was on a test and I could not do!!</p>
| lab bhattacharjee | 33,337 | <p>$$\binom{p-1}k=\frac{(p-1)\cdots(p-k)}{k!}=\prod_{r=1}^k\frac{p-r}r$$</p>
<p>Now $p-r\equiv-r\pmod p\implies\dfrac{p-r}r\equiv-1$</p>
|
923,000 | <p>I am confused because I have seen implies and equivalent used interchangibly. For instance, I've seen </p>
<p>$$x-y=0 \implies x=y$$</p>
<p>And I've also seen</p>
<p>$$x-y=0 \Longleftrightarrow x=y$$</p>
<p>Are both of these statements correct? Which one am I supposed to use?</p>
<p>I know that implies is tr... | Tara Nanda | 754,648 | <p>I have found the 3 volumes by Garling "A Course in Mathematical Analysis" published by Cambridge University Press excellent. The author has several years of teaching experience and has thought through the subject in great depth.
Vol. I ISBN: 978-1-107-03202-6
Vol. II ISBN: 978-1-107-03203-3
Vol III ISBN: 978-1-107-... |
1,001,839 | <p>$$\frac{\pi x y^2}{4}$$</p>
<p>Is this function continuous? I really haven't worked with continuity with multivariable funtions before, so I am a little stumped. How would one answer such a question? </p>
<p>I'm reading a bit ahead of my level, and I'm seeing all these epsilon delta things... is that what I am sup... | Vladimir Vargas | 187,578 | <p>Continuity in $\mathbb R^2$ of a function is defined as follows:</p>
<p>Let $X$ be an open subset of $\mathbb R^2$ and $f$ a real valued function from $X \subseteq \mathbb R^2 \rightarrow \mathbb R$. We say that $f$ is continuous over $X$ if $\forall (x_0,y_0)\in X\wedge\forall\varepsilon>0\;\exists\,\delta >... |
2,832,311 | <p>Suppose I draw 10 tickets at random with replacement from a box of tickets, each of which is labeled with a number. The average of the numbers on the tickets is 1, and the SD of the numbers on the tickets is 1. Suppose I repeat this over and over, drawing 10 tickets at a time. Each time, I calculate the sum of the n... | Mikhail Katz | 72,694 | <p>To understand the definition and be able to work with it properly it may be helpful to use an equivalent definition that has one variable less. Namely, a sequence $(a_n)$ converges to $a$ if for every $\epsilon>0$ there is an $N\in \mathbb N$ such that all the terms
$$
a_N, a_{N+1}, a_{N+2}, \ldots
$$
are within... |
149,049 | <p>Suppose you have a list of intervals (or tuples), such as:</p>
<pre><code>intervals = {{3,7}, {17,43}, {64,70}};
</code></pre>
<p>And you wanted to know the intervals of all numbers not included above, e.g.:</p>
<pre><code>myRange = 100;
numbersNotUed[myRange,intervales]
(*out: {{1,2},{8,16},{44,63},{71,100}}*)... | eldo | 14,254 | <pre><code>int = {{3, 7}, {17, 43}, {64, 70}};
com = Partition[#, 2]& @ {1, Sequence @@ Riffle[int[[All, 1]] - 1, int[[All, 2]] + 1], 100}
</code></pre>
<blockquote>
<p>{{1, 2}, {8, 16}, {44, 63}, {71, 100}}</p>
</blockquote>
<pre><code>NumberLinePlot[{Interval @@ int, Interval @@ com}, PlotTheme -> "Detail... |
3,541,897 | <p>While searching for non-isomorph subgroups of order <span class="math-container">$2002$</span> I just encountered something, which I want to understand. Obviously I looked for abelian subgroups first and found <span class="math-container">$2002=2^2*503$</span> so we have the groups
<span class="math-container">$$
\m... | user729424 | 729,424 | <p>First let's note that <span class="math-container">$2^2\cdot503=2012\ne2002$</span>.</p>
<p><strong>Abelian groups of order 2002:</strong></p>
<p>There is only one Abelian group of order <span class="math-container">$2002$</span>, namely</p>
<p><span class="math-container">$$\Bbb{Z}_{2002}\cong\Bbb{Z}_{2}\times\B... |
844,700 | <p>I am looking for a calculator which can calculate functions like $f(x) = x+2$
at $x=a$ etc; but I am unable to do so. Can you recommend any online calculator?</p>
| Ned | 521,624 | <p>You can use the Calcpad online calculator for free:
Just go to <a href="http://calcpad.net/Calculator" rel="nofollow noreferrer">http://calcpad.net/Calculator</a>
Try to type the following example into the 'Script' box:</p>
<pre><code>f(x) = x + 2
a = 4
f(a)
a = 6
f(a)
$Plot{f(x) @ x = 0 : a}
</code></pre>
<p>Then... |
510,151 | <p>Prove by induction that $2k(k+1) + 1 < 2^{k+1} - 1$ for $ k > 4$.
Can some one pls help me with this?</p>
<p>I reformulated like this</p>
<p>$ 2k(k+1) + 1 < 2^{k+1} - 1 $</p>
<p>$ 2k^2+2k+2<2^{k+1}$</p>
<p>and I tried like this
Take $k=k+1$</p>
<p>$ 2^{k+2} -1 > 2(k+1)(k+2) + 1 $</p>
<p>$... | Julián Aguirre | 4,791 | <p>Because in the discrete metric the only converging sequences are the constant sequences from some point on.</p>
<p>In the case of $\{1/n\}\subset\mathbb{R}$ with the discrete metric $d$ we have
$$
d(1/n,0)=1.
$$</p>
|
4,369,232 | <p>I have the following problem:</p>
<blockquote>
<p>Let <span class="math-container">$\{(M_i,\tau_i)\}_{i\in I}$</span> be nonempty topological spaces where <span class="math-container">$I$</span> is arbitrary but non empty. Let <span class="math-container">$M=\prod_{i\in I} M_i$</span>. Let <span class="math-containe... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$O$</span> be a subbasic element in the product i.e. <span class="math-container">$O=\pi_i^{-1}[U]$</span> for some <span class="math-container">$i \in I$</span> and some open <span class="math-container">$U \subseteq X_i$</span> (I use <span class="math-container">$\pi_i$</span> for... |
104,335 | <p>I am implementing a code that works correctly but it takes too much time. I did not see how to optimize it to be run quickly. Here is my code:</p>
<pre><code>data=RandomInteger[{1,400},{5000,2}];
c=10;
r=60;
pts=c + r {Cos[#], Sin[#]} & /@ Range[0, 2 π, 2 π/16];
newCoord = Table[(# - pts[[i]]) & /@ data,... | Diogo | 36,260 | <p>If you use float instead integer you can reduce the computing time.</p>
<pre><code> data = RandomInteger[{1, 400}, {5000, 2}];
c = 10.;
r = 60.;
pts = c + r {Cos[#], Sin[#]} & /@ Range[0., 2. π, 2. π/16.];
newCoord = Table[(# - pts[[i]]) & /@ data, {i, 1, Length@pts}];
PolarCoords = ... |
2,007,403 | <p>Determine the convergence or divergence of </p>
<p>$$\sqrt[n]{n!}$$</p>
<p>I was trying to use the propriety $\lim_{n\to\infty}\sqrt[n]{n}=1$, maybe I can write this</p>
<p>$\lim_{n\to\infty}\sqrt[n]{n!}=\lim_{n\to\infty}\sqrt[n]{n \ \times(n-1)\times(n-2)\times(n-3)\cdots2\ \times \ 1}=\lim_{n\to\infty}\sqrt[n]n... | zhw. | 228,045 | <p>Hint: If $n$ is even, then $n!> (n/2)^{n/2}.$</p>
|
34,487 | <p>A few years ago Lance Fortnow listed his favorite theorems in complexity theory:
<a href="http://blog.computationalcomplexity.org/2005/12/favorite-theorems-first-decade-recap.html" rel="nofollow">(1965-1974)</a>
<a href="http://blog.computationalcomplexity.org/2006/12/favorite-theorems-second-decade-recap.html" rel=... | Stefan Geschke | 7,743 | <p>My favourite results are (1) the existence of NP-complete problems (Cook), (2) the Baker-Gil-Solovay theorem that whether P=NP holds relativized to on oracle depends on the oracle,
and (3) Fagin's characterization of NP in terms of second order logic.</p>
<p>I am not so much interested in the large number of proofs... |
1,116,009 | <p>Suppose $\alpha$,a,b are integers and $b\neq-1$. Show that if $\alpha$ satisfies the equation $x^2+ax+b+1=0$,then prove $a^2+b^2$ is composite.</p>
<p>I am starting with this study course of polynomials and finding it very difficult to understand. Please help me with the question. Thanks in advance ! </p>
| mollyerin | 29,809 | <p>I wrote a lot, due to boredom and the general hope that some of it is helpful. Much will probably be familiar to you.</p>
<p>First: In order to work with tensors or to do calculus on manifolds at all, it's very important to start making the distinction between <em>vectors</em> and <em>covectors</em> (or "dual vecto... |
1,783,200 | <p>Prove or disprove the following statement:</p>
<p><strong>Statement.</strong> <em>Continuous for each variables, when other variables are fixed, implies continuous?</em> More clearly, prove or disprove the following problem:</p>
<p>Let $\displaystyle f:\left[ a,b \right]\times \left[ c,d \right]\to \mathbb{R}$ for... | Robert Israel | 8,508 | <p>Standard example:
$$ f(x,y) = \dfrac{xy}{x^2 + y^2}$$
with $f(0,0) = 0$.</p>
|
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | Gab | 15,326 | <p>If you want to evaluate the polynomial at a lot of equidistant points, you can do "forward differencing"; here are 3 slides explaining the method: <a href="http://zach.in.tu-clausthal.de/teaching/info2_11/folien/evaluating%20a%20polynomial%20at%20equidistant%20points.pdf" rel="nofollow">http://zach.in.tu-clausthal.d... |
43,886 | <p>Has work been done on looking at what happens to the exponents of the prime factorization of a number $n$ as compared to $n+1$? I am looking for published material or otherwise. For example, let $n=9=2^0\cdot{}3^2$, then,</p>
<p>$$
9 \;\xrightarrow{+1}\; 10
$$</p>
<p>$$
2^0\cdot{}3^2 \;\xrightarrow{+1}\; 2^1\cd... | JavaMan | 6,491 | <p>I found this on MO which is relevant to the question:</p>
<p><a href="https://mathoverflow.net/questions/55010/prime-factorization-of-n1">https://mathoverflow.net/questions/55010/prime-factorization-of-n1</a></p>
|
2,225,606 | <p>Solution: The eigenvalues for $\begin{bmatrix}1.25 & -.75 \\ -.75 & 1.25\end{bmatrix}$ are $2$ and $0.5$. </p>
<p>I'm confused on how it's not $1$ and $-1$. If we set up the characteristic matrix: $\begin{bmatrix}5/4 - \lambda & -3/4 \\ -3/4 & 5/4 - \lambda \end{bmatrix}$ </p>
<p>$ad-bc=0$</p>
<p>... | mrnovice | 416,020 | <p>$$(\frac{5}{4}-\lambda)^2-\frac{9}{16}=0$$</p>
<p>Assuming you got the above equation correctly, your expansion of the terms was incorrect.</p>
<p>$$\frac{25}{16}-\frac{5}{2}\lambda+\lambda^2-\frac{9}{16}=0$$</p>
<p>$$\lambda^2-\frac{5}{2}\lambda+1=0$$</p>
<p>$$(\lambda-\frac{5}{4})^2+\frac{-25+16}{16}=0$$</p>
... |
122,728 | <p>Suppose that I have a <a href="http://en.wikipedia.org/wiki/Symmetric_matrix" rel="nofollow noreferrer">symmetric</a> <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix" rel="nofollow noreferrer">Toeplitz</a> <span class="math-container">$n\times n$</span> matrix</p>
<p><span class="math-container">$$\mathbf{A}=\... | john316 | 262,158 | <p>Define some variables for convenience
$$\eqalign{
P &= {\rm Diag}(\beta) \cr
B &= cP^{-1} \cr
b &= {\rm diag}(B) \cr
S &= A+B \cr
M &= AS^{-1}A \cr
}$$
all of which are symmetric matrices, except for $b$ which is a vector.</p>
<p>Then the function and its differential can be expressed... |
1,639,568 | <p>The above applies $\forall x,y \in \mathbb{R}$</p>
<p>I've tried: $x + y \ge 0$</p>
<p>$$x + y \ge x$$</p>
<p>$$ (x + y)^2 \ge 2xy$$</p>
<p>$$\frac{(x + y)^2}{2} \ge xy$$</p>
<p>But the closest I get is $\dfrac{x+y}{\sqrt{2}} \ge \sqrt{xy}$</p>
<p>Any ideas?</p>
| L.F. Cavenaghi | 248,387 | <p>$$(x-y)^2 \ge 0$$
$$x^2 - 2xy + y^2 \ge 0 $$
$$x^2 + y^2 \ge 2xy $$
$$x^2 + 2xy + y^2 \ge 4xy $$
$$(x+y)^2 \ge 4xy $$</p>
|
1,014,303 | <blockquote>
<p>Is
$$\sum^\infty_{n=4}\frac{3^{2n}}{(-10)^n}$$
Convergent or Divergent? Explain why.</p>
</blockquote>
<p>I know I can do:
$$\sum^\infty_{n=4}\frac{9^{n}}{(-10)^n} \Rightarrow \sum^\infty_{n=4}\bigg(\frac{9}{-10}\bigg)^n$$
But I'm not sure where to go from here. The negative denominator is really... | Pedro | 23,350 | <p>We're assuming $R$ is a ring in which every prime ideal is f.g., so $I$ being non f.g. <em>should be non prime</em> (this is the "Observe $I$ is not prime" line). This guarantees the existence of such $J_1,J_2$ as in the proof in the post. The point is that, as proved, any such ideal <em>is prime</em>. Inevitably we... |
3,573,811 | <p>This is a theorem given by my professor from Artin Algebra:</p>
<p>Suppose that a finite abelian group <span class="math-container">$V$</span> is a direct sum of cyclic groups of prime orders <span class="math-container">$d_j=p_j^{r_j}$</span>. The integers <span class="math-container">$d_j$</span> are uniquely det... | Captain Lama | 318,467 | <p>This is the first time I see this argument, I find it rather amusing. Let's look at it in detail.</p>
<p>Suppose <span class="math-container">$G$</span> is a direct sum of groups of the type <span class="math-container">$\mathbb{Z}/p^i\mathbb{Z}$</span>. Write it as
<span class="math-container">$$G = \mathbb{Z}/p\m... |
2,280,243 | <blockquote>
<p>A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its <em>seeds</em> For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $... | Γιώργος Πλούσος | 422,616 | <p>I don't understand exactly what the question is, but the following result may be useful (I do not quote the proof):</p>
<p><strong>Tribonacci</strong></p>
<p>The formula for calculating the nth term is equivalent to the following relationships where only one cubic constant is used instead of three.</p>
<p><span clas... |
2,280,243 | <blockquote>
<p>A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its <em>seeds</em> For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $... | Community | -1 | <p>By the theory of linear recurrences, the sequence approximately follows a geometric progression</p>
<p>$$u_n=ar^n$$ where $r$ is the largest root of $r^3=r^2+r+1$, which is about $1.8392867552142$.</p>
<p>With $u_7=292$, we estimate $a=4.1005$.</p>
<p>Then we can expect $u_n\ge10000$ for </p>
<p>$$n\ge\frac{\log... |
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