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 |
|---|---|---|---|---|
3,913,005 | <p>So say there is 26 possible characters and you've got 1800 long random string. Now how would you go about finding the chance of there being a specific 5 letter word within?</p>
<p>I already found a related question to this but the given formula in the answer doesn't seem to make sense: <a href="https://math.stackexc... | user2661923 | 464,411 | <p>This is a long-winded comment that is being posted as an answer only for legibility.</p>
<p>The only approach that I am aware of is <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferrer">Inclusion-Exclusion</a> which results in very ugly math.</p>
<p>I will make the s... |
398,371 | <p>How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get</p>
<p>$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightarrow1^+}\frac{2\pi\cos(\pi t)\sqrt{1+\cos(\pi t)}}{-\pi\sin(\pi t)}$$... | egreg | 62,967 | <p>I'll show you a trick: since $t\to 1^+$ you know that $\pi t\to \pi^+$, so $(\pi t -\pi)\to 0^+$.</p>
<p>First of all do the change of variable $u=\pi t-\pi$, so you get $\pi t=u-\pi$. Then $\sin(\pi t)=\sin(u-\pi)=-\sin u$ and $\cos(\pi t)=\cos(u-\pi)=-\cos u$. Thus your limit becomes</p>
<p>$$
\lim_{u\to 0^+}\fr... |
878,686 | <p>How do I derive the $m$ in the formula:
$$I=\left(1+\frac{r}{m}\right)^{mn} -1$$</p>
<p>all the values of the variables in the formula except $m$ is given and the question is find $m$.
I just don't know how to derive the formula using the knowledge of Algebra I have.</p>
| Community | -1 | <p>Using the answer by David, we set $t=\dfrac rm$, and</p>
<p>$$\sqrt[nr]{1+I}=(1+t)^{1/t}=s,$$
or
$$1+t=s^t=e^{t\ln(s)}.$$</p>
<p>Then
$$(1+t)s=e^{(1+t)\ln(s)},$$
$$-(1+t)\ln(s)\,e^{-(1+t)\ln(s)}=-\frac{\ln(s)}s,$$
and
$$-(1+t)\ln(s)=W(-\frac{\ln(s)}s).$$
Hence,
$$m=\frac r{-\dfrac{W(-\frac{\ln(s)}s)}{\ln(s)}-1}.$... |
64,096 | <p>I'm trying to define a function that is evaluated differently depending on its
options. This code</p>
<pre><code>Options[Foo]={Bar->True};
Foo[a_,b_, OptionsPattern[]]:= A /; OptionValue[Bar];
Foo[a_,b_, OptionsPattern[]]:= B /; !OptionValue[Bar];
</code></pre>
<p>works as expected. Evaluating </p>
<pre><code... | rcollyer | 52 | <p><strong>Note</strong>: This is an incomplete analysis and leads to the wrong conclusion about the cause of the difficulty. Mr.W's <a href="https://mathematica.stackexchange.com/a/64193/52">answer</a> below correctly identifies the culprit as <code>Condition</code>.</p>
<hr>
<p>The problem you are facing has nothin... |
64,096 | <p>I'm trying to define a function that is evaluated differently depending on its
options. This code</p>
<pre><code>Options[Foo]={Bar->True};
Foo[a_,b_, OptionsPattern[]]:= A /; OptionValue[Bar];
Foo[a_,b_, OptionsPattern[]]:= B /; !OptionValue[Bar];
</code></pre>
<p>works as expected. Evaluating </p>
<pre><code... | Mr.Wizard | 121 | <p>Alright, I took another look at this issue and I do <em>not</em> believe this is a duplicate of:</p>
<ul>
<li><a href="https://mathematica.stackexchange.com/q/1567/121">How can I create a function with "positional" or "named" optional arguments?</a></li>
</ul>
<p>However I also do not believe t... |
3,140,194 | <p>how to solve a system equation with radical</p>
<p><span class="math-container">$$\sqrt{x+y}+\sqrt{x+3}=\frac{1}{x}\left(y-3\right)$$</span></p>
<p>And <span class="math-container">$$\sqrt{x+y}+\sqrt{x}=x+3$$</span></p>
<p>This system has <span class="math-container">$1$</span> root is <span class="math-container... | TheSilverDoe | 594,484 | <p><span class="math-container">$f(x)^{g(x)} = \exp(g(x) \ln(f(x))$</span>, so if <span class="math-container">$f(x)$</span> has a positive finite limit <span class="math-container">$a$</span>, and <span class="math-container">$g(x)$</span> has a finite limit <span class="math-container">$b$</span>, then by continuity ... |
3,140,194 | <p>how to solve a system equation with radical</p>
<p><span class="math-container">$$\sqrt{x+y}+\sqrt{x+3}=\frac{1}{x}\left(y-3\right)$$</span></p>
<p>And <span class="math-container">$$\sqrt{x+y}+\sqrt{x}=x+3$$</span></p>
<p>This system has <span class="math-container">$1$</span> root is <span class="math-container... | Hagen von Eitzen | 39,174 | <p>Not always.</p>
<p>We have <span class="math-container">$0^0=1$</span>, but <span class="math-container">$\lim_{x\to x_0} f(x)=\lim_{x\to x-0} g(x)=0$</span> does not imply <span class="math-container">$\lim_{x\to x_0}f(x)^{g(x)}=1$</span>. The problem is that the exponentiation map <span class="math-container">$(x... |
2,775,087 | <blockquote>
<p>Without using a calculator, what is the sum of digits of the numbers from $1$ to $10^n$?</p>
</blockquote>
<p>Now, I'm familiar with the idea of pairing the numbers as follows:</p>
<p>$$\langle 0, 10^n -1 \rangle,\, \langle 1, 10^n - 2\rangle , \dotsc$$</p>
<p>The sum of digits of each pair is $9n$... | deficiencyOn | 532,135 | <p>Okay so the truly simple explanation is merely the fact that all those pairs sum to $999$ (i.e. in the case 0f $n=3$) . There's nothing more than that. In more details, one can see that this property is true for the first pair. Inductively, the next pair is made by increasing the first item by $1$ and decreasing the... |
2,775,087 | <blockquote>
<p>Without using a calculator, what is the sum of digits of the numbers from $1$ to $10^n$?</p>
</blockquote>
<p>Now, I'm familiar with the idea of pairing the numbers as follows:</p>
<p>$$\langle 0, 10^n -1 \rangle,\, \langle 1, 10^n - 2\rangle , \dotsc$$</p>
<p>The sum of digits of each pair is $9n$... | Love Invariants | 551,019 | <p>There is a constant pattern of nos. I will demonstrate it for 2 digit nos.<br>
00 01 02 03 04 05 06 07 08 09<br>
10 11 12 13 14 15 16 17 18 19<br>
20 21 22 23 24 25 26 27 28 29 and it goes so on. </p>
<p>You can make a pattern of numbers for any digit nos.<br>
Just add $0s$ before a number to make it of n-digit. N... |
368,800 | <p>Let <span class="math-container">$G$</span> be a finite group, let <span class="math-container">$X$</span> be a locally compact Hausdorff space, and let <span class="math-container">$G$</span> act freely on <span class="math-container">$X$</span>. It is well-known that the canonical quotient map <span class="math-co... | Hannes Thiel | 24,916 | <p>Let <span class="math-container">$X=[-1,1]^\infty\setminus\{0\}$</span>, which is a metrizable, locally compact space. Consider the two-element group <span class="math-container">$G$</span>, and the free <span class="math-container">$G$</span>-action on <span class="math-container">$X$</span> given by <span class="m... |
1,752,506 | <p>Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$</p>
<p>My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$</p>
<p>$=> (x^2 + 9) - \sqrt{x^2 + 9} \sqrt{x^2 + 1} = 0$</p>
<p>$=> \sqrt{x^2 + 9} (\sqrt{x^2 + 9} - \sqrt{x^... | Merkh | 141,708 | <p>Yes it is correct. It would do harm dividing out $\sqrt{x^2 + 9}.$ Perhaps an easier example of why this is so... Consider
\begin{equation}
x^2 = x.
\end{equation}
Obviously the two roots are 0,1, but do you see what happens when you divide by $x?$ You reduce the order of the polynomial, hence throwing away a root... |
2,482,564 | <p>Let $(\Omega,\Sigma)$ be a given measurable space and let $f$ be a $\Sigma$-measurable function. If $h:[-\infty,\infty]\rightarrow[-\infty,\infty]$ is a continuous function, then the composite function $hf$ is measurable.</p>
<p>This can be proven easily if every continuous function is measurable as I just need to ... | Gono | 384,471 | <p>Usually with "measurable" on the real line "Borel measurable" is meant... if you do also every continuous function is ofc measurable because the preimage of open sets under continuous functions are open sets as well and open sets are a generator of the Borel Algebra</p>
|
11,922 | <p>Which is correct terminology: "A Cartesian plane" or "The Cartesian plane"? (As in the directions for a section of homework being, "Plot a point on ______ Cartesian plane." In that context, I feel that one of the two should be used consistently, but find there's little agreement on Google for which is appropriate.</... | Gerald Edgar | 127 | <p>Why do you think they should be used consistently? I think there are situations appropriate for each of these choices.</p>
|
653,449 | <p>According to <a href="http://en.wikibooks.org/wiki/Haskell/Category_theory" rel="noreferrer">the Haskell wikibook on Category Theory</a>, the category below is not a valid category due to the addition of the morphism <em>h</em>. The hint says to " think about associativity of the composition operation." Bu... | Martin Brandenburg | 1,650 | <p>From the graph we see that $f$ <em>and</em> $h$ are inverse to $g$. Now it is a general fact about categories that inverses of morphisms are unique. The proof is the same as the one for groups.</p>
|
2,066,501 | <p>If Q = \begin{bmatrix}1&1/√6&1/3√3\\1&-1/√6&-1/3√3\\0 & √2/√3 & -1/3√3\end{bmatrix}</p>
<p>and R = \begin{bmatrix}2√2&1/√2&1/√2\\0&4/√6&1/√6\\0&0&1/3√3\end{bmatrix}.</p>
<p>Is A invertible? (no computation required) Is the system Ax=b solvable for each b in $R^3$ (gi... | Dave | 334,366 | <p>Since $A=QR$ we have: $\det(A)=\det(QR)=\det(Q)\det(R)$</p>
<p>Since $Q$ is orthogonal (i.e. $Q^{-1}=Q^T$), we have: $1=\det(I)=\det(QQ^T)=\det(Q)\det(Q^T)=(\det(Q))^2$ so $\det(Q)=\pm 1\neq 0$.</p>
<p>Since $R$ is upper triangular, and has all nonzero elements on its diagonal, $\det(R)\neq 0$.</p>
<p>Thus $\det(... |
662,744 | <h2>Question Statement</h2>
<blockquote>
<p>Let $K$ be a finite field of $q$ elements. Let $U$, $V$ be vector spaces over $K$ with
$\dim(U) = k$, $\dim(V) = l$. How many linear maps $U \rightarrow V$ are there?</p>
</blockquote>
<p>I'm struggling to answer this question. Given that a linear map is uniquely determ... | Sugata Adhya | 36,242 | <p><strong>Hint:</strong> The space of all linear maps from $U\to V$ is isomorphic to $\text{Mat}_{l\times k}(K).$</p>
|
410,663 | <p>Can anyone in detail explain the procedure for finding the harmonic conjugate of the function $$u(x,y) = x^{2} - y\cdot (y+1)$$</p>
<p>I am new to this and I would like to know. </p>
| ˈjuː.zɚ79365 | 79,365 | <p>Following Peter Tamaroff's solution (integration of $v_y=u_x=2x$ and $v_x=-u_y=2y+1$) one obtains $v(x,y)=2xy+f(x)$ and $v(x,y)=2xy+x+g(y)$, from where $v(x,y)=2xy+x+c$ where $c$ is a constant. This is the general form of harmonic conjugate. </p>
<p>With some fluency in complex arithmetics, you can solve the proble... |
268,461 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/228080/operatornameimfz-leq-operatornamerefz-then-f-is-constant">$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant</a> </p>
</blockquote>
<blockquote>
<p>Let $f\colon\mathbb C \t... | Robert Israel | 8,508 | <p>Every member of $K$ is either in $F$ or in $F^c$. If it's in $F$, it's covered by the open cover of $F$. If it's in $F^c$, it's covered by $F^c$. So in either case, it's covered. </p>
|
2,361,275 | <p>$$\neg(A\land B)\land(B\lor\neg C)\Leftrightarrow(\neg A\land B)\lor(\neg B\land\neg C)$$
Checked their truth tables are the same, but can you show the steps how to transform one to the other?</p>
| Peter | 463,556 | <p>We have:
$$\overline{AB} (B \vee \bar{C}) \Leftrightarrow (\bar{A} \vee \bar{B})(B \vee \bar{C}) \Leftrightarrow \bar{A}B \vee \bar{A} \bar{C} \vee \bar{B} B \vee \bar{B} \bar{C} \Leftrightarrow \bar{A}B \vee \bar{A} \bar{C} \vee \bar{B} \bar{C}$$
Now if $\bar{A} \bar{C}$ turns out to be true, either $\bar{A} B$ or ... |
1,655,884 | <blockquote>
<p>How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$?</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p>This seems like a hard question, since I can't even think of one example to ... | poetasis | 546,655 | <p>For your equation, solutions include
<span class="math-container">$$(1,1,1)\quad (1,2,2)\quad (1,3,3)\quad (1,3,3)\quad
(1,4,4)\quad (1,5,5)\quad \cdots $$</span>
Any triple with zero or one for one of <span class="math-container">$\space x,y\space $</span> will work. All others would probably require a brute force... |
182,301 | <p>I have asked a question <a href="https://mathematica.stackexchange.com/questions/182247/constructing-possibilities-of-a-generating-function">here</a>.
I want to reproduce the coefficients of a generating function of the form:
<span class="math-container">$$(1 + x)^2 (1 + x + x^2 + x^3+\cdots+x^n)^{n-1}$$</span>
It ... | Carl Woll | 45,431 | <p><strong>Update</strong></p>
<p>An even faster method (I also modified the order of each possibility as requested in the comments):</p>
<pre><code>tups[n_] := Values @ GroupBy[
Tuples[Join[{Range[2]}, ConstantArray[Range[n+1], n-1], {Range[2]}] - 1],
Total
]
</code></pre>
<p>A comparison with the accepted ... |
627,575 | <p>There is a whiskey made up of 64% corn, 32% rye, and 4% barley that was made by blending other whiskies together. I am trying to figure out if there is a chance the ratio of this whiskey could be the result of blending two, maybe three whiskies of different ratios.</p>
<p>The possible whiskies:</p>
<p>Whiskey A is... | MJD | 25,554 | <p><strong>Hint</strong> Consider the three base whiskies as three unit vectors over the three-dimensional vector space of corn, rye, and barley. Mixing whiskeys then corresponds to simple operations on vectors, and you should be able to express this mixing operation in terms of operations you have already studied. Th... |
4,594,043 | <p>For <span class="math-container">$|x|<1,$</span> we have
<span class="math-container">$$
\begin{aligned}
& \frac{1}{1-x}=\sum_{k=0}^{\infty} x^k \quad \Rightarrow \quad \ln (1-x)=-\sum_{k=0}^{\infty} \frac{x^{k+1}}{k+1}
\end{aligned}
$$</span></p>
<hr />
<p><span class="math-container">$$
\begin{aligned}
\int... | Leucippus | 148,155 | <p>Starting with the integral
<span class="math-container">$$ \int_{0}^{1} (1-x)^m \, x^{-t} \, dx = B(m+1, 1-t), $$</span>
a case of the Beta function, then
<span class="math-container">\begin{align}
\partial_{m} \int_{0}^{1} (1-x)^m \, x^{-t} \, dx &= \partial_{m} \, B(m+1, 1-t) \\
\int_{0}^{1} \ln(1-x) \, (1-x)^... |
2,287,878 | <p>Given the following polynomial: $$P(x)=(x^2+x+1)^{100}$$ How do I find : $$\sum_{k=1}^{200} \frac{1}{1+x_k} $$ Is there a general solution for this type of problem cause I saw they tend to ask the same thing for $\sum_{k=1}^{200} \frac{1}{x_k}$? Also how do I find the coefficient of $a_1$ and the remainder for $$P(x... | florence | 343,842 | <p>Note that the roots of $x^2+x+1$ are
$$\zeta_3 =e^{2\pi i/3} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}$$
and
$$\zeta_3^2 =e^{4\pi i/3} = -\frac{1}{2}-i\frac{\sqrt{3}}{2}$$
Both of these are roots of $P(x)$ with multiplicity $100$. Therefore,
$$\sum_{k=1}^{200}\frac{1}{1+x_k} = 100\left( \frac{1}{1+\zeta_3}+\frac{1}{1+\ze... |
3,757,972 | <p>If <span class="math-container">$\frac{ab} {a+b} = y$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are greater than zero, why is <span class="math-container">$y$</span> always smaller than the smallest number substituted?</p>
<p>Say <span class="math-container">$a... | Peter | 82,961 | <p>We have <span class="math-container">$$\frac{ab}{a+b}<\frac{ab}{a}=b$$</span> and <span class="math-container">$$\frac{ab}{a+b}<\frac{ab}{b}=a$$</span> if <span class="math-container">$a,b>0$</span></p>
|
3,757,972 | <p>If <span class="math-container">$\frac{ab} {a+b} = y$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are greater than zero, why is <span class="math-container">$y$</span> always smaller than the smallest number substituted?</p>
<p>Say <span class="math-container">$a... | J. W. Tanner | 615,567 | <p>If <span class="math-container">$a,b>0$</span> then <span class="math-container">$a<a+b$</span> and <span class="math-container">$\dfrac a{a+b}<1$</span> so <span class="math-container">$\dfrac {ab}{a+b}<b$</span>.</p>
<p>A similar argument shows that <span class="math-container">$\dfrac{ab}{a+b}<a$</... |
3,757,972 | <p>If <span class="math-container">$\frac{ab} {a+b} = y$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are greater than zero, why is <span class="math-container">$y$</span> always smaller than the smallest number substituted?</p>
<p>Say <span class="math-container">$a... | DMcMor | 155,622 | <p>Well, so long as <span class="math-container">$a,b > 0$</span> it's certainly true that <span class="math-container">$$ab < aa + ab,$$</span> but that gives us <span class="math-container">$$ab < a(a+b) \implies \frac{ab}{a+b} < a.$$</span></p>
<p>Similarly, <span class="math-container">$$ab < ab + bb... |
3,007,785 | <p><span class="math-container">$\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$</span> I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, there is very difficult calculations. I have the answer <span class="math-container... | Community | -1 | <p>There is a simple way to address the denominator:</p>
<p><span class="math-container">$$(x+\tan x-\sin2x)'=1+\tan^2x+1-2\cos 2x=\tan^2x+4\sin^2x=\sin^2x\left(\frac1{\cos^2x}+4\right).$$</span></p>
<p>The second factor will tend to <span class="math-container">$5$</span> and the first can be replaced by <span class... |
2,874,840 | <blockquote>
<p>If $P\left(A\right)=0.8\:$ and $P\left(B\right)=0.4$, find the maximum and minimum values of $\:P(A|B)$.</p>
</blockquote>
<p>My textbook says the answer is $0.5$ to $1$. But I think the answer should be $0$ to $1$.</p>
<p>The textbook claims $P(A∩B)$ is $0.2$ when $P(A'∩B')=0$</p>
<p>I think that ... | StubbornAtom | 321,264 | <p>Since $P(A\cap B)$ is less than both $P(A)$ and $P(B)$,</p>
<p>we have $$P(A\cap B)\le \min(P(A),P(B))=0.4$$ And from Bonferroni's inequality it follows that $$P(A\cap B)\ge \max(0,P(A)+P(B)-1)=0.2$$</p>
<p>Now, $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A\cap B)}{0.4}$$</p>
<p>So, $$0.5\le P(A\mid B)\le 1$$</p... |
11,290 | <p>I have this code to produce an interactive visualization of a tangent plane to a function:</p>
<pre><code>Clear[f]
f[x_, y_] := x^3 + 2*y^3
Manipulate[
Show[
Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, PlotStyle -> Opacity[0.8]],
Plot3D[f[point[[1]], point[[2]]] +
Limit[(f[point[[1]] + h, point[[2]]] - f[po... | Mark McClure | 36 | <p>Simply move the derivative computation outside the scope of the <code>Manipulate</code>, using either <code>D</code> or <code>Limit</code>.</p>
<pre><code>f[x_, y_] = x^3 + 2 y^3;
fx[x_, y_] = D[f[x, y], x];
fx[x_, y_] = Limit[(f[x + h, y] - f[x, y])/h, h -> 0];
fy[x_, y_] = D[f[x, y], y];
fy[x_, y_] = Limit[(f[... |
338,099 | <p>Are there general ways for given rational coefficients <span class="math-container">$a,b,c$</span> (I am particularly interested in <span class="math-container">$a=3,b=1,c=8076$</span>, but in general case too) to answer whether this equation has a rational solution or not?</p>
| Xarles | 24,442 | <p>Your curve is a genus 1 curve, usually expressed as
<span class="math-container">$$ y^2= -abx^4+bc$$</span>
(just by multiplying everything by <span class="math-container">$b$</span> and changing <span class="math-container">$y$</span> by <span class="math-container">$by$</span>). </p>
<p>The curve has local point... |
2,299,678 | <p>Question:</p>
<p>Assume $x, y$ are elements of a field $F$. Prove that if $xy = 0$, then $x = 0$ or $y = 0$.</p>
<p>My thinking:</p>
<p>I am not sure how to prove this. <strong>I can only use basic field axioms.</strong> Should I assume that both x and y are not equal to 0 and then prove by contradiction or shoul... | CY Aries | 268,334 | <p>Hint:</p>
<p>If $x=0$, then we are done.</p>
<p>If $x\ne 0$, then $x^{-1}$ exists. Multiply $x^{-1}$ to both sides of $xy=0$.</p>
|
101,078 | <p>While reading through several articles concerned with mathematical constants, I kept on finding things like this:</p>
<blockquote>
<p>The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. </p>
<p>The <strong>high-water marks</strong> are ... | Bill Dubuque | 242 | <p>Truncating a continued fraction right before a "high-water" (large) partial quotient $\rm\: a_{i+1}\:$ yields a particularly good rational approximation since </p>
<p>$$\rm r\ =\ [a_0;\ a_1;\ a_2;\ \cdots\ ]\ \ \Rightarrow\ \ \left|\ r\ -\ \frac{p_i}{q_i}\:\right|\ \le\: \frac{1}{a_{i+1}\:q_i^2}$$</p>
<p>For exam... |
3,756,932 | <p>Overall, I think I understand the Monty Hall problem, but there's one particular part that I either don't understand, or I don't agree with, and I'm hoping for an intuitive explanation why I'm wrong or confirmation that I'm right.</p>
<p>The variant that I'm working on is the one with 3 doors. Prize is uniformly loc... | Barry Cipra | 86,747 | <p>If Monty tells you <em>in advance</em> that, after you've made your initial selection, he will open one of the other two doors, revealing it to be empty, and then give the option of switching to the door he didn't open, he is, in effect, telling you that you can either stick with what's behind your initial selection... |
3,756,932 | <p>Overall, I think I understand the Monty Hall problem, but there's one particular part that I either don't understand, or I don't agree with, and I'm hoping for an intuitive explanation why I'm wrong or confirmation that I'm right.</p>
<p>The variant that I'm working on is the one with 3 doors. Prize is uniformly loc... | zkutch | 775,801 | <p>I prefer following explanation:</p>
<p>From scratch I have probability <span class="math-container">$\frac{1}{3}$</span> to win and <span class="math-container">$\frac{2}{3}$</span> against me. Because Monty should open door with empty choice, then that <span class="math-container">$\frac{2}{3}$</span> which was aga... |
220,946 | <p>I am a newcomer to Mathematica and I am trying to solve this differential equation:</p>
<pre><code>s = NDSolve[{y'[x] == -4/3*y[x]/x + 4 a/(3 x^2) + 4/(9 x^3) a b -
8/(9 x^3) c, y[0.01] == 20}, y, {x, 0.01, 10}]
</code></pre>
<p>I am getting the error message "Encountered non-numerical value for a derivativ... | Cesareo | 62,129 | <p>A "brute force" solution with Genetic Algorithms </p>
<p>Given a symbolic matrix, first we convert to a zero-one's matrix in which the ones represent non null elements. This is done as follows. Given <strong>M</strong> we obtain <strong>M0</strong></p>
<pre><code>{n, n} = Dimensions[M]
M0 = Table[If[NumericQ[M[[i,... |
1,414,690 | <p>This is a followup to my question <a href="https://math.stackexchange.com/questions/1414636/eigenvalues-of-matrix-with-all-1s">here</a>.</p>
<p>Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. Is $A$ diagonalizable?</p>
| user1551 | 1,551 | <p>$A$ is always diagonalisable when the field has characteristic 0 or characteristic $>n$.</p>
<p>Let $e_i$ denotes the vector with a $1$ at the $i$-th position and zeros elsewhere. Let $P$ be the matrix whose first and second columns are respectivelly $\sum_je_j$ and $e_1-e_2$ and whose $j$-th column is $e_2-e_j$... |
69,187 | <p>Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of course be obtained if $A$, $B$ have a common eigenspace of multiplicity at least $m$.</p>
<p>My question is: is it ... | Igor Rivin | 11,142 | <p>If you restrict to the postulated $m$-dimensional eigenspace $E$ of $C=xA + yB,$ the fact that $C+\epsilon A$ has the same multiplicity, means that $A$ restriced to $E$ is also a multiple of identity, as is $B$ (by the same argument). Since a small perturbation of a matrix with distinct eigenvalues still has disti... |
69,187 | <p>Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of course be obtained if $A$, $B$ have a common eigenspace of multiplicity at least $m$.</p>
<p>My question is: is it ... | David E Speyer | 297 | <p>This is false. Let $A_0$ and $B_0$ be $k \times k$ symmetric matrices with no common eigenspace. Let $A$ and $B$ be the $2k \times 2k$ matrices with block forms $\left( \begin{smallmatrix} A_0 & 0 \\ 0 & A_0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} B_0 & 0 \\ 0 & B_0 \end{smallmatri... |
69,187 | <p>Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of course be obtained if $A$, $B$ have a common eigenspace of multiplicity at least $m$.</p>
<p>My question is: is it ... | Denis Serre | 8,799 | <p>I think that the most interesting examples come from systems of conservation laws in physics. These are systems of PDEs (partial differential equations). When they are first-order and linear (linearity can be achived by linearizing about a constant state), you have a system
$$\partial_tU+\sum_{\alpha=1}^dA^\alpha U=... |
2,291,175 | <p>I know that the domain is: $(-\infty,0) \cup (0,\infty)$, or all Reals except Zero.</p>
<p>But if a take the "nearest negative number to zero" and then, the "nearest positive number to zero", my function will hugely increase (Y will approach +infinity).</p>
<p>And my domain have all numbers that approach Zero in b... | James S. Cook | 36,530 | <p>The question is how you define <strong>increasing function</strong> ? Suppose we define increasing function as one for which $a,b \in \text{dom}(f)$ with $a\leq b$ implies $f(a)\leq f(b)$. Then, for $f(x) = 1/x$ for $x \in \mathbb{R} - \{ 0 \} = (-\infty,0) \cup (0, \infty)$ we certainly <strong>cannot</strong> clai... |
2,716,036 | <p>In reviewing some old homework assignments, I found two problems that I really do not understand, despite the fact that I have the answers.</p>
<p>The first is: R(x, y) if y = 2^d * x for some nonnegative integer d. What I do not understand about this relation is how it can possibly be transitive (according to my n... | José Carlos Santos | 446,262 | <ol>
<li>If $x\mathop Ry$ and $y\mathop Rz$, then there are non-negative integers $d$ and $d'$ such that $y=2^dx$ and $z=2^{d'}y$. Therefore, $z=2^{d+d'}x$ and, since $d+d'$ is a non-negative integer, $x\mathop Rz$.</li>
<li>No, it is not reflexive. For instance, $1\not\mathop R1$.</li>
</ol>
|
2,900,283 | <p>I'm having trouble proving this theorem from Rudin: that if $x \neq 0$ then $\frac{1}{\frac{1}{x}} = x$. </p>
<p>Rudin seems to solve this by referring to an earlier result that $xy = xz$ for $x \neq 0$ implies that $y = z$. I haven't quite been able to grasp this approach, as we don't seem to have access to this a... | Peter Szilas | 408,605 | <p>Suggestion:</p>
<p>Let $x\not =0.$</p>
<p>$x^{-1}= 1/x.$</p>
<p>Starting with the second line:</p>
<p>$ (xx^{-1})(x^{-1})^{-1}= $</p>
<p>$x(x^{-1}(x^{-1})^{-1})=x \cdot 1=x$.</p>
|
1,248,329 | <p>Let $R$ be an integral domain, and let $r \in R$ be a non-zero non-unit. Prove that $r$ is irreducible if and only if every divisor of $r$ is either a unit or an associate of $r$.</p>
<p>Proof. ($\leftarrow$) Suppose $r$ is reducible then $r$ can be expressed as $r = ab$ where $a$, $b$ are not units. This contradic... | Stefan Hamcke | 41,672 | <p>You are right about the first part. If $p\omega$ is nullhomotopic in $A$, then the homotopy to the constant loop lifts to a homotopy between $\omega$ and the constant loop. This is because of the lifting properties of covering maps. Note that a covering map $p:\tilde X\to X$ restricts to a covering map $p^{-1}(A)\to... |
510,488 | <p>Using the fact that $x^2 + 2xy + y^2 = (x + y)^2 \ge 0$, show that the assumption $x^2 + xy + y^2 < 0$ leads to a contradiction...
So do I start off with...</p>
<p>"Assume that $x^2 + xy + y^2 <0$, then blah blah blah"?</p>
<p>It seems true...because then I go $(x^2 + 2xy + y^2) - (x^2 + xy + y^2) \ge 0$.
It... | bof | 97,206 | <p>Averaging the inequalities $(x+y)^2\ge0$ and $x^2+y^2\ge0$ we get $\frac12(x+y)^2+\frac12(x^2+y^2)\ge0$, that is $x^2+xy+y^2\ge0$, contradicting the assumption that $x^2+xy+y^2\lt0$.</p>
|
2,071,828 | <p>Find the splitting field of $x^6-2x^4-8x^2+16$ over $\mathbb {F}_3$ and list the intermediate fields between the base camp and the splitting field.</p>
| GRE | 378,788 | <p>My factorization was incorrect, I have understood my error. Now to calculate the Galois group of the extension (which is actually a Galois extension, as $\mathbb {F}_9$ is the splitting field of a separable polynomial, having first derivative different from zero, with coefficients in $\mathbb {F}_3$) I just note tha... |
1,619,679 | <p>If the series $\sum\limits_{n=1}^{\infty} u_n $ is convergent then the sequence $u_n \rightarrow 0$ as $n \rightarrow \infty$. Therefore if the ratio test $R=\frac{u_{n+1}}{u_n}$ gives $R<1$ then we can conclude that $(u_{n})_{n\in \mathbb{N}}$ is convergent, right?</p>
<p>Generally, if we can find that a series... | Ben Grossmann | 81,360 | <p>Yes, you have the right idea. Whenever
the series $\sum_{n=1}^\infty u_n$ converges, the sequence $u_n$ must converge to zero as $n \to \infty$. This is true <em>even if</em> the sum is alternating. Certainly, then: if $\sum u_n$ converges by the ratio test, then we must have $u_n \to 0$ as $n \to \infty$.</p>
<... |
2,928,806 | <p><span class="math-container">$m(t)=\frac{t^2}{1-t^2},t<1$</span></p>
<p>The suggested answer provided by our teacher states that m(0)=0, so there's no real value with this function as mgf; hence mgf doesn't exist.</p>
<p>Intuitively I understand that <span class="math-container">$e^{xt}>0$</span> should alwa... | J.G. | 56,861 | <p>For any mgf <span class="math-container">$M$</span>, <span class="math-container">$M(0)=1$</span>. Since <span class="math-container">$m(0)=0\ne 1=M(0)$</span>, <span class="math-container">$m$</span> is not an mgf.</p>
|
1,004,056 | <p>Pick a piece of paper and a pen. Put the pen on a starting point and begin to draw an arbitrary curve and don't withdraw your hand until you reached the starting point. You can meet your curve during drawing in some <em>one point</em> intersections but not more (like a line intersection). For example the left curve ... | John Hughes | 114,036 | <p>If the curve is smooth enough, and there are finitely many self-intersections, the answer is yes. For this answer, I'm going to limit myself to a particular class of curves, though: those where at most <em>two</em> bits of curve intersect at any crossing, so you can have something that locally looks like $\times$, b... |
33,743 | <p>I have a lot of sum questions right now ... could someone give me the convergence of, and/or formula for, $\sum_{n=2}^{\infty} \frac{1}{n^k}$ when $k$ is a fixed integer greater than or equal to 2? Thanks!!</p>
<p>P.S. If there's a good way to google or look up answers to these kinds of simple questions ... I'd lo... | Steven Stadnicki | 785 | <p>It's also easy to show that this series converges through more elementary means (though without getting a good estimate of the value), by a version of the same elementary technique often used to show that the harmonic series diverges. Start by fixing $k=2$; since the terms for any other $k$ are smaller than this, t... |
854,671 | <p>So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. </p>
<p>For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$.
So I kind of understand how to center the circle and solve this with polar coordinates. Since the circle equation is $(x-3)^2... | Michael Hardy | 11,667 | <p>Just brute force:</p>
<p>There is a calendar for a common year (i.e. a non-leap year) beginning on Sunday.</p>
<p>There is a calendar for a common year beginning on Monday.</p>
<p>There is a calendar for a common year beginning on Tuesday.</p>
<p>. . . and so on. Seven calendars. Then seven more for leap years... |
3,429,350 | <p>If we a sample of <span class="math-container">$n$</span> values from a given population and if <span class="math-container">$X$</span> is the variable of the sample, then the mean of <span class="math-container">$X$</span> is just <span class="math-container">$\dfrac{ \sum x }{n}$</span></p>
<p>Now, suppose <span ... | Randy Marsh | 391,136 | <p>It is equal to one because it contains <span class="math-container">$K_5$</span> and we can exhibit an embedding in genus 1. In general the problem of determining the genus of a graph is NP-hard.</p>
<p>Here is an approach for finding an embedding in orientable genus 1:</p>
<p>Cut out two open disks from the spher... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | William M. | 396,761 | <p><strong>Hint</strong>. Use that <span class="math-container">$\lim (1 + \frac{x}{n})^n \to e^x$</span> for any real number <span class="math-container">$x.$</span></p>
|
708,633 | <p>My question is as in the title: is there an example of a (unital but not necessarily commutative) ring $R$ and a left $R$-module $M$ with nonzero submodule $N$, such that $M \simeq M/N$?</p>
<p>What if $M$ and $N$ are finitely-generated? What if $M$ is free? My intuition is that if $N$ is a submodule of $R^n$, then... | egreg | 62,967 | <p>A classical example is the Prüfer $p$-group $\mathbb{Z}(p^{\infty})$ ($p$ a prime); for each proper subgroup $H$ of it, it holds $\mathbb{Z}(p^{\infty})\cong \mathbb{Z}(p^{\infty})/H$ (and there's plenty of subgroups).</p>
<p>If $R$ is the endomorphism ring of an infinite dimensional vector space, then, as (left) m... |
932,430 | <p>I am studying about how a real number is defined by its properties. The three type of properties that make the real numbers what they are.</p>
<ol>
<li><p>Algebraic properties i.e, the axioms of addition, subtraction multiplication and division.</p></li>
<li><p>Order properties i.e., the inequality properties</p></... | Ben | 33,679 | <p>Suppose I have a set $X \subseteq \mathbb{R}$ such that for all $x \in X$ we have $x \le t$ for some number $t$. Then the completeness axiom guarantees the existence of a smallest number, call it $s$, such that $x \le s$.</p>
<p>You write this down more formally by saying that if $X$ is bounded, <em>ie.</em> if all... |
348,674 | <p>I am a beginner at matrices and I am trying to find out whether or not the linear transformation defined by the matrix $A$ is onto, and also whether it is 1-1.</p>
<p>Here is the matrix $A$: </p>
<p>$$\begin{bmatrix}
1 & 3 & 4 & -1 & 2\\
2 & 6 & 6 & 0 & -3\\
3 &... | Dimitri | 62,474 | <p>If the reduced matrix have a zero column, then it is not 1-1 because for example $A(00001)^{T}$ will be zero, this will be the case, beacuse the matrix $A$ that a vector in $\mathbb{R}^5$ to $\mathbb{R}^4$, so it can´t be 1-1.
To see if it is into, you have to see that the columns generate $\mathbb{R}^4$, so if the ... |
1,672,882 | <p>Why we always see extended operations, like arbitrary unions, products, etc. in different parts of mathematics in the form of extensions of finite ones upon arbitrary <em>sets</em> (called index set)? Why never use an index-<em>class</em> for a proper class?</p>
<p>EDIT:
Of course I think this a reason, for example... | hmakholm left over Monica | 14,366 | <p>The restrictions on what you can use proper classes for are there to make sure that everything you write down can actually be proved to exist using the restricted set-building axioms of ZF set theory. We can prove once and for all that a union or product (or whatever) indexed by a <em>set</em> will always describe s... |
108,297 | <p>I've been told that strong induction and weak induction are equivalent. However, in all of the proofs I've seen, I've only seen the proof done with the easier method in that case. I've never seen a proof (in the context of teaching mathematical induction), that does the same proof in both ways, and I can't seem to f... | Patrick Da Silva | 10,704 | <p>The idea is that if something is proved with "strong" induction, i.e. by assuming all preceding cases, then you can use "weak" induction on the hypothesis "all preceding cases hold". Let me explain with mathematical notation, perhaps it'll be a little clearer.</p>
<p>Suppose you want to prove a proposition for all ... |
3,902,501 | <p>Over <span class="math-container">$\mathbb R$</span>, the only linear maps are those of the form <span class="math-container">$ax$</span>.</p>
<p>If we discuss rational functions over <span class="math-container">$\mathbb R$</span>, this extra structure would allow us to describe a wider variety of linear maps.</p>
... | Robert Israel | 8,508 | <p>Any linear map <span class="math-container">$T$</span> of vector spaces <span class="math-container">$X \to Y$</span> can be "generalized" to a map of
<span class="math-container">$Z \to Y$</span>, where <span class="math-container">$Z$</span> is any vector space that contains <span class="math-container"... |
2,574,962 | <p>An even graph is a graph all of whose vertices have even degree. </p>
<p>A spanning subgraph $H$ of $G$, denoted by $H \subseteq_{sp} G$, is a graph obtained by $G$ by deleting <em>only</em> edges of $G$.</p>
<p>I want to show that if $G$ is a connected graph, then
$\big|\{H \subseteq_{sp} G | H$ $is$ $even\}\big|... | Yly | 253,140 | <p>Pick a base vertex $v$. For any other vertex $w$, pick a path $P(v,w)$ from $v$ to $w$. $P(v,w)$ exists because $G$ is connected. </p>
<p>For a spanning subgraph $H\subseteq_{sp} G$, we speak of "flipping an edge $e$" to mean removing $e$ from $H$ if $e\in H$, and adding $e$ to $H$ if $e$ is not in $H$. This tr... |
713,626 | <p>If $X$ is $Beta\left(\dfrac{ \alpha_1}{ 2 }, \dfrac{\alpha_2}{2}\right)$ then $\dfrac{\alpha_2 X}{\alpha_1(1-X)}$ is $F(\alpha_1, \alpha_2)$? </p>
<p>Any help is appreciated I don't know where to start. I'm assuming I need the pdf's of each distribution?</p>
| Frazer | 125,581 | <p>Yes, I use the pdf.</p>
<p>$f(x)=\frac{\Gamma(\frac{m}{2}+\frac{n}{2})}{\Gamma(\frac{m}{2})\Gamma(\frac{n}{2})}x^{\frac{m}{2}-1}(1-x)^{\frac{n}{2}-1}$</p>
<p>and the pdf of function $g(x)$ is $l(y)=f(h(y))|h'(y)|$, where $h$ is the inverse of $g$. </p>
<p>We get $h(y)=\frac{my}{my+n}$ and $|h'(y)|=\frac{mn}{(my+n... |
1,615,732 | <p>Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, so we have $\widetilde H_i(X)=0$ for all $i$. And if $\pi_2(X)=0$, we have that $X$ is $K(\pi,1)$.</p>
<p>My questio... | Hari Rau-Murthy | 142,843 | <p>$\newcommand{\Z}{\mathbb{Z}}$
The statement $\pi_2(X)=0$ would imply that $X$ is a $K(\pi,1)$, is not true. I will construct for you now an acyclic space,(the space $X$ below) with $\pi_2=0$, that is not an eilenberg maclane space.</p>
<p>Let $2I$ be the $\Z/2$ extension of the icosahedral group, $I$, given by the... |
119,686 | <p>I have a very dumb question. Let $X = \mathbb{P}^2_k = Proj(k[x,y,z])$ where $k$ is algebraically closed. We have an invertible sheaf $\mathcal{O}(2)$ on $X$. Its space of global sections contains the elements $x^2, y^2, z^2, xy, yz, xz$. </p>
<p>It seems to me (by my calculations), however, that $\mathcal{O}(2... | rfauffar | 12,158 | <p>The thing is, with $x^2$, $y^2$ and $z^2$, there's no way of generating $xy$, $xz$ and $yz$; that's the problem. That's why these necessarily have to be part of the generating set.</p>
|
1,874,159 | <blockquote>
<p>Given real numbers $a_0, a_1, ..., a_n$ such that $\dfrac {a_0}{1} + \dfrac {a_1}{2} + \cdots + \dfrac {a_n}{n+1}=0,$ prove that $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n=0$ has at least one real solution.</p>
</blockquote>
<p>My solution:</p>
<hr>
<p>Let $$f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a... | Marco Cantarini | 171,547 | <p>You can also prove it using the mean value theorem. You showed that $$\int_{0}^{1}f\left(x\right)dx=0
$$ now since $f$ is continuous by the <a href="https://en.wikipedia.org/wiki/Mean_value_theorem#First_Mean_Value_Theorem_for_Definite_Integrals">mean value theorem for integrals</a> we have that exists some $c\in\l... |
3,702,309 | <p>The question reads,
Find all differentiable functions <span class="math-container">$f$</span> such that <span class="math-container">$$f(x+y)=f(x)+f(y)+x^2y$$</span> for all <span class="math-container">$x,y \in \mathbb{R} $</span>. The function <span class="math-container">$f$</span> also satisfies <span class="ma... | nonuser | 463,553 | <p>First notice by letting <span class="math-container">$x=y=0$</span> we get <span class="math-container">$f(0)=0$</span>. Further we have <span class="math-container">$$f(x+y)-f(x) = f(y) +x^2y$$</span> now if <span class="math-container">$y\ne 0$</span> we have <span class="math-container">$$\lim _{y\to 0} {f(x+y)-f... |
471,561 | <p><img src="https://ukpstq.bn1.livefilestore.com/y2p5St1yRZbdxBvzMbBTYjqjqtwDvBaoWtc7YRGZXCwBTax0XseUIh_l_O92NO6XAbLeGaqU67bkBI4lroIlcD2ade_rxfDast52B_7ECcMd68/question3.png?psid=1" alt="question"></p>
<p>I have attempted these few simple questions, can someone let me know if this is correct please? If not please pro... | dtldarek | 26,306 | <p>That's not quite right. Here's a <strong>hint</strong>:</p>
<ul>
<li>What are the necessary suffixes for $L_1$ and $L_2$?</li>
<li>Suppose $w_1 \in L_1$. Does $\mathtt{a}w_1$ or $\mathtt{b}w_1$ belong to $L_1$?</li>
<li>Suppose $w_2 \in L_2$. Does $\mathtt{a}w_2$ or $\mathtt{b}w_2$ belong to $L_2$?</li>
</ul>
<p>I... |
4,623,748 | <p>Let <span class="math-container">$X,Y$</span> be smooth vector fields on the unit circle <span class="math-container">$M = S^1$</span> such that <span class="math-container">$[X,Y] = 0$</span>, i.e. (treating tangent vectors as derivations) <span class="math-container">$X(x)(Yf) = Y(x)(Xf)$</span> for all <span clas... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$z^2+|z|=0$</span>, then <span class="math-container">$z^2=-|z|\leqslant0$</span>. The only complex numbers whose square is a real number smaller than or equal to <span class="math-container">$0$</span> are those numbers <span class="math-container">$z$</span> of the form <span class=... |
1,851,698 | <p>While playing with the results of defining a new operation, I came across a number of interesting properties with little literature surrounding it; the link to my original post is here: <a href="https://math.stackexchange.com/questions/1785715/finding-properties-of-operation-defined-by-x%E2%8A%95y-frac1-frac1x-frac1... | Mitchell Spector | 350,214 | <p>$$f_n = (\phi_i^{\;-n}-\phi_i^{\;n})\frac{i}{\sqrt{3}}$$</p>
<p>You can prove this by induction on $n,$ using the fact that $\phi_i$ and $\phi_i^{\;-1}$ are the two solutions to the quadratic equation $x^2-x+1=0.$ (Alternatively, it's easy to see that the three sequences $\langle \phi_i^{\;n}\;\vert\;n \in \mathbb... |
1,099,214 | <p>I'm having trouble calculating some probabilities for a simple card game I'm into </p>
<p>1) So say we have a deck of 13 cards total. You draw 3 cards so there are 10 cards left in the deck. Say there is 1 special card in the deck that we want to figure out the probability of getting it in the hand. How do we figur... | drhab | 75,923 | <p>Hint on 2)</p>
<p>$P(\text{the special AND}\geq1 \text{ copies of...})=P(\text{the special})-P(\text{the special AND 0 copies of...})$</p>
|
238,673 | <p><a href="http://vihart.com/doodling/" rel="nofollow">Vi Hart's doodling videos</a> and a 4 year old son interested in mazes has made me wonder:</p>
<p>What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria:</p>
<p>1) They are "solitaire" games, i.e. require only one ... | Rosie F | 344,044 | <p>Or run a <a href="https://en.wikipedia.org/wiki/Turmite" rel="nofollow">turmite</a> which moves on a grid of square cells, and never "decreases" the colour of a cell (where this colour is viewed as a number). Represent cell-colours as e.g. $\Box, \boxminus, \boxplus, \blacksquare$ (where every cell starts as $\Box$... |
19,478 | <p>Let $K$ and $L$ be two subfields of some field. If a variety is defined over both $K$ and $L$, does it follow that the variety can be defined over their intersection?</p>
| Bjorn Poonen | 2,757 | <p>No, not even for genus $0$ curves, if "<strong>$X$ is defined over $K$</strong>" means that the variety $X$, initially defined over an extension $F$ of $K$, is to be isomorphic to the base extension of some variety over $K$. (Pete in his answer was implicitly assuming that he was allowed to base extend to an algebr... |
973,985 | <p>I don't know where to start. </p>
<p>I know that the sum of the angles is less than or equal to 180.
but how do i prove this.</p>
| Michael Hardy | 11,667 | <p>It seems what must have been intended is this: Find <em>one</em> triangle in which it is easy to prove that the sum is $180^\circ$.</p>
<p>Then deduce that if it's the same in all other triangles, then it must be $180^\circ$ in all triangles.</p>
<p><b>PS:</b> OK, let's look at the isosceles right triangle. The <... |
973,985 | <p>I don't know where to start. </p>
<p>I know that the sum of the angles is less than or equal to 180.
but how do i prove this.</p>
| Sherlock Holmes | 173,514 | <p>Draw a generic triangle. Produce a line through a vertex parallel to the opposite side. Applying alternate angles, the result is seen. </p>
|
1,038,060 | <p>Can anyone help me with this question: I know it before, but I have tried to solve it myself and didnt succeed.
what is the regular expression for this language: L=all words that have 00 or 11 but not both.</p>
<p>Thank you!</p>
| Aaron Maroja | 143,413 | <p>As suggested, I am writing the complete answer.</p>
<p>First we use the not so simple result: </p>
<p><strong>Theorem:</strong> Let $n = 3$ or $n\geq 5$. Then $A_n$ is simple. </p>
<p><strong>Statement:</strong> Now for $n=3$ or $n \geq 5$ let's show that $\{id\}, A_n$ and $S_n$ are the only normal subgroups of $... |
1,038,060 | <p>Can anyone help me with this question: I know it before, but I have tried to solve it myself and didnt succeed.
what is the regular expression for this language: L=all words that have 00 or 11 but not both.</p>
<p>Thank you!</p>
| user1729 | 10,513 | <p>The following idea is super-simple. However, it only works if $n$ is odd (so it works precisely half the time!). But as it is so simple I thought it would be nice to record it here.</p>
<p>Suppose $n$ is odd. Note that $S_n$ can be generated by the elements $\alpha:=(1, 2)$ and $\beta:=(1, 2, \ldots, n)$. Now, ever... |
3,672,839 | <p>What is the derivative of <span class="math-container">$f(x)=|x|^\frac{3}{2},\forall x\in \mathbb{R}$</span>.</p>
<p>When <span class="math-container">$x>0$</span> it is fine but the problem is when <span class="math-container">$x\leq 0$</span>.I was trying by the defintion of derivative but what is <span class=... | Community | -1 | <p>We were hit with Zorn's lemma in the first group theory course before any other (this is in India). I never had the pleasure of taking courses in logic and set theory, and ring theory naturally came only later, so this answer is technically different from Asaf's.</p>
<blockquote>
<p><em>Exercise.</em> Let <span c... |
136,808 | <p>Let $A$ be a nonempty subset of $\omega$, the set of natural numbers.
I want to prove this statement:</p>
<blockquote>
<p>If $\bigcup A=A$ then $n\in A \implies n^+\in A$.</p>
</blockquote>
<p>Help...</p>
| hmakholm left over Monica | 14,366 | <p>$\bigcup A\subseteq A$ says that $A$ is transitive and is therefore an ordinal. Now if $A$ were a successor ordinal $\alpha+1 = \alpha\cup\{\alpha\}$, then $\alpha\in A$ but $\bigcup A = (\bigcup\alpha)\cup\alpha \not\ni \alpha$. Thus $A$ must be either $0$ or $\omega$.</p>
<hr>
<p>Or more directly: Assume $n\in A... |
2,867,718 | <p>How do I integrate $\sin\theta + \sin\theta \tan^2\theta$ ?</p>
<p>First thing,
I have been studying maths for business for approximately 3 months now. Since then, I studied algebra and then I started studying calculus. Yet, my friend stopped me there, and asked me to study Fourier series as we'll need it for our ... | PM. | 416,252 | <p>$$
\sin\theta+\sin\theta\tan^2\theta=\sin\theta(1+\tan^2\theta)=\sin\theta\sec^2\theta=\sec\theta\tan\theta
$$
because $(1+\tan^2\theta)=\sec^2\theta$ is a 'standard' identity.</p>
<p>Also
$$
\frac{d\sec\theta}{d\theta}=\sec\theta\tan\theta
$$
is a 'standard' derivative.</p>
<p>Therefore
$$
\int(\sin\theta+\sin\th... |
1,905,308 | <p>In the book "A Course in Metric Geometry" (By Dmitri Burago, Yuri Burago Sergei Ivanov), there is a short proof of lower semicontinuity of length induced by a metric: (Prop 2.3.4, <a href="https://books.google.com/books?id=dRmIAwAAQBAJ&pg=PA35" rel="nofollow noreferrer">pg 35</a>)</p>
<p>Let $\gamma_j:[a,b] \to... | David C. Ullrich | 248,223 | <p>This is not an answer to your specific question about the proof in the book. It seems worthwhile to point out that the lower semicontinuity of arc length is much more trivial than that proof makes it appear - the proof you sketch seems to me to be sort of missing the point.</p>
<p>Since $\sum_j(Y)$ is just a finit... |
765,738 | <p>I am trying to prove the following inequality:</p>
<p>$$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$</p>
<p>for all $a>0, b>0$.</p>
<p>Does anyone know how to prove it?</p>
<p>Thanks a lot in advance!</p>
| egreg | 62,967 | <p>It's not restictive to assume $a>b$, so we can write $a=e^{2s}$, $b=e^{2t}$, with $s>t$ and the inequality to prove becomes
$$
4(e^s-e^t)^2\le(e^s-e^t)(e^s+e^t)(2s-2t)
$$
or
$$
\frac{e^s-e^t}{s-t}\le\frac{e^s+e^t}{2}
$$
By Lagrange's theorem, we know that
$$
\frac{e^s-e^t}{s-t}=e^u
$$
for some $u$, $t<u<... |
2,359,408 | <p>Question:</p>
<p>Let $\\f: \ \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ via $ f(x) = (x+2, x-3)$. Is $f$ injective? Is $f$ surjective?</p>
<p>I was able to prove that $f$ is injective. However, I am not quite sure if $f$ is surjective. If it is surjective could someone please tell me how to prove that. If not, c... | Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>Let $y=f(x)=( y_1 , y_2 ),$ notice that $y_1-y_2=5$</p>
<p>Now consider point $(6,0)$, does it have a preimage?</p>
|
2,359,408 | <p>Question:</p>
<p>Let $\\f: \ \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ via $ f(x) = (x+2, x-3)$. Is $f$ injective? Is $f$ surjective?</p>
<p>I was able to prove that $f$ is injective. However, I am not quite sure if $f$ is surjective. If it is surjective could someone please tell me how to prove that. If not, c... | freakish | 340,986 | <p>Well, you need to show that for any $(a,b)\in\mathbb{R}^2$ there exists $x\in\mathbb{R}$ such that $f(x)=(a,b)$. Since $f(x)=(x+2, x-3)$ then $a=x+2$ and $b=x-3$. In particular $x=a-2$ and $x=b+3$ and therefore $a-2=b+3$ and finally $a=b+5$. So that's the constraint on $(a,b)$ in order for it to be in the image of $... |
548,563 | <p>Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$</p>
<p>I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$</p>
<p>I have no idea about it. :|</p>
| Bruno Joyal | 12,507 | <p>Recall the Euler Beta integral</p>
<p>$$\beta(a, b) = \int_0^1 x^{a-1} (1-x)^{b-1} dx$$</p>
<p>and Euler's formula for it in terms of the Gamma function,</p>
<p>$$\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$$</p>
<p>In particular, since ${2n \choose n} = (2n!)/(n!)^2$, we have</p>
<p>$$\beta(n+1, n+1... |
1,867,352 | <p>Find all the angles $v$ between $-\pi$ and $\pi$ such that</p>
<p>$$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$</p>
<p>The answer has to be in the form of: $\pi/2$ (it must include $\pi$)</p>
<p>I have tried squaring but I get nowhere.</p>
| bthmas | 222,365 | <p>As noted by another user, it's useful to combine a sum of sines and cosines into a single function.
$$
\sqrt{3}\cos(v) - \sin(v) = \sqrt{2}
$$
In order to turn the LHS into a single function we have a bit of a choice here we can make the single function a sine or a cosine, and we can also choose if we want an additi... |
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | BenPen | 365,278 | <p>I can see a superset proof: If you transform it into $2^{2n} + 1$, it's a well known proof that shows it's not divisible by 3. (maybe the original answer gave it away too much...)</p>
|
245,623 | <p><em>For the following vectors $v_1 = (3,2,0)$ and $v_2 = (3,2,1)$, find a third vector $v_3 = (x,y,z)$ which together build a base for $\mathbb{R}^3$.</em></p>
<p>My thoughts:</p>
<p>So the following must hold:</p>
<p>$$\left(\begin{matrix}
3 & 3 & x \\
2 & 2 & y \\
0 & 1 & z
\end{matrix}\... | Gerry Myerson | 8,269 | <p>The big mistake is at the very beginning --- there is no reason at all why you should want that equation to hold. </p>
<p>There are infinitely many correct choices for $v_3$. One simple one is the cross product of $v_1$ and $v_2$ (warning --- this choice won't be available in other vector spaces). </p>
|
245,623 | <p><em>For the following vectors $v_1 = (3,2,0)$ and $v_2 = (3,2,1)$, find a third vector $v_3 = (x,y,z)$ which together build a base for $\mathbb{R}^3$.</em></p>
<p>My thoughts:</p>
<p>So the following must hold:</p>
<p>$$\left(\begin{matrix}
3 & 3 & x \\
2 & 2 & y \\
0 & 1 & z
\end{matrix}\... | Adam Rubinson | 29,156 | <p>Well (3,2,0) and (3,2,1) give you (0,0,1). So for your third one (0,1,0) would work.</p>
<p>Then (1,0,0) = 1/3 [(3,2,0) - 2(0,1,0)]</p>
|
1,207,247 | <blockquote>
<p>Prove that if $\gcd(a,b)=1$, then $\gcd(a^2+b^2, a^2b^2)=1$.</p>
</blockquote>
<p>My attempt:</p>
<p>If $a$ is prime to $b$ the $gcd(a,b)=1$. Assume that $a^2+b^2$ and $a^2b^2$ are not prime to each other. Let $d=gcd(a^2+b^2, a^2b^2)$. Then $d|a^2+b^2, ~~ \&~~d|a^2b^2$. We shall have to prove th... | Awais Chishti | 223,303 | <p>We have $d|a^2(a^2+b^2)-a^2b^2$, $d|a^4$ and by symmetry, $d|b^4$. So $d|gcd(a^4,b^4)=gcd(a,b)^4=1$</p>
|
577,163 | <p>Let $A=(a_{ij})_{n\times n}$ such
$a_{ij}>0$ and $\det(A)>0$.</p>
<p>Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$.</p>
<p>This problem is from my friend, and I have considered sometimes, but I can't. Thank you </p>
| user1551 | 1,551 | <p>The statement is false. Counterexample:
$$
B=\pmatrix{2&2&2\\ 3&4&2\\ 3&1&4},\ A=B\circ B\circ B=\pmatrix{8&8&8\\ 27&64&8\\ 27&1&64},\ \det(B)=-2,\ \det(A)=7000.
$$</p>
<p><strong>Edit.</strong> However, if $B$ is positive definite, the <em>converse</em> of your state... |
1,545,045 | <p>Is it possible to cut a pentagon into two equal pentagons?</p>
<p>Pentagons are not neccesarily convex as otherwise it would be trivially impossible.</p>
<p>I was given this problem at a contest but cannot figure the solution out, can somebody help?</p>
<p>Edit: Buy "cut" I mean "describe as a union of two pentag... | Graham Kemp | 135,106 | <p>Since gender doesn't restrict anything, and people are unique individuals, we just need to count ways to divide twenty-eight people into fourteen pairs.</p>
<p>There are $28!$ ways to line everyone up, then split into $14$ pairs. However, each pair can be formed in $2!$ ways, and we don't care about the $14!... |
1,668,487 | <p>$$2^{-1} \equiv 6\mod{11}$$</p>
<p>Sorry for very strange question. I want to understand on which algorithm there is a computation of this expression. Similarly interested in why this expression is equal to two?</p>
<p>$$6^{-1} \equiv 2\mod11$$</p>
| 5xum | 112,884 | <p>It's because $2\cdot 6 \equiv 1 \mod 11$</p>
|
2,683,788 | <blockquote>
<p>An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$
balls bear the same number $i$ for each $i=1,2,...n$. $N$ balls are drawn at random without replacement. Find the probability that exactly $m$ of the numbers will appear in the sample.</p>
</blockquote>
<p>Any hints would be grea... | drhab | 75,923 | <p>Think of it as if the socks are picked up one by one.$$P(bbb)+P(bbg)+P(bgb)+P(gbb)=$$$$\frac{18}{34}\frac{17}{33}\frac{16}{32}+\frac{18}{34}\frac{17}{33}\frac{16}{32}+\frac{18}{34}\frac{16}{33}\frac{17}{32}+\frac{16}{34}\frac{18}{33}\frac{17}{32}=$$$$\frac{18}{34}\frac{17}{33}\frac{16}{32}+3\cdot\frac{18}{34}\frac{1... |
2,294,585 | <p>I can't figure out why $((p → q) ∧ (r → ¬q)) → (p ∧ r)$ isn’t a tautology. </p>
<p>I tried solving it like this: </p>
<p>$$((p ∧ ¬p) ∨ (r ∧ q)) ∨ (p ∧ r)$$
resulting in $(T) ∨ (p ∧ r)$ in the end that should result in $T$. What am I doing wrong?</p>
| DanielV | 97,045 | <p>$$((p \Rightarrow q) \land (r \Rightarrow ¬q)) \Rightarrow (p ∧ r)$$</p>
<p>If it is January, then I am cold.</p>
<p>If it is July, then I am not cold.</p>
<p>Therefore, it is January and July.</p>
<p>Would you accept that?</p>
|
2,473,132 | <p>Compute $$ \int_{0}^{ \infty }\frac{\ln xdx}{x^{2}+ax+b}$$
I tried to compute the indefinite integral by factorisation and partial fraction decomposition but it became nasty pretty soon. There must be another way to directly evaluate it without actually computing the indefinite integral which I don't know!</p>
| pisco | 257,943 | <p>In order for your integral to converge, we must have $b>0$.</p>
<p>Hence
$$\begin{aligned}\int_0^\infty {\frac{{\ln x}}{{{x^2} + ax + b}}dx} &= \sqrt b \int_0^\infty {\frac{{\ln (\sqrt b x)}}{{b{x^2} + a\sqrt b x + b}}dx} \\ &= \sqrt b \ln \sqrt b \int_0^\infty {\frac{1}{{b{x^2} + a\sqrt b x + b}}dx... |
4,013,638 | <p>a) <span class="math-container">$F(x)>0$</span> for every <span class="math-container">$x>0$</span>.<br>
b) <span class="math-container">$F(x)>F(e)$</span> for every <span class="math-container">$x>1$</span>.<br>
c) <span class="math-container">$x=e^{-2}$</span> is an inflection point. <br>
d) None of th... | VIVID | 752,069 | <ul>
<li>(<span class="math-container">$a$</span>) is wrong because <span class="math-container">$F(1) = 0$</span>.</li>
<li>(<span class="math-container">$b$</span>) is wrong because <span class="math-container">$F'(x)=x^{\sqrt {x}} > 0$</span> so the function is increasing.</li>
<li>(<span class="math-container">$... |
86,657 | <p>Throughout my upbringing, I encountered the following annotations on Gauss's diary in several so-called accounts of the history of mathematics:</p>
<blockquote>
<p>"... A few of the entries indicate that the diary was a strictly private affair of its author's (sic). Thus for July 10, 1796, there is the entry</p>
... | Kristal Cantwell | 1,098 | <p>I believe I have a reference for Schuhmann:</p>
<p>Schuhmann E. 1976 Vicimus GEGAN, Interpretationsvarianten zu einer Tagebuchnotiz von C.F. Gauss, Naturwiss. Tech. Medezin. 13.2, 17-20.</p>
|
1,139,723 | <p>Given the differential equation $\frac{dy}{dt}+a(t)y=f(t)$ </p>
<p>with a(t) and f(t) continuous for:</p>
<p>$-\infty<t<\infty$</p>
<p>$a(t) \ge c>0$</p>
<p>$lim_{t_->0}f(t)=0$</p>
<p>Show that every solution tends to 0 as t approaches infinity.</p>
<p>$$\frac{dy}{dt}+a(t)y=f(t)$$</p>
<p>$$μ(t)=e^... | Victor | 213,782 | <p>After your last step:</p>
<p>$$e^{\frac{1}{2}a^2(t)}y=\int e^{\frac{1}{2}a^2(t)}[f(t)]dt$$</p>
<p>You want to divide both sides by $e^{\frac{1}{2}a^2(t)}$ to solve for $y$</p>
<p>Giving us:</p>
<p>$$y=e^{-\frac{1}{2}a^2(t)}\int e^{\frac{1}{2}a^2(t)}[f(t)]dt$$</p>
|
3,129,852 | <p>My question is pretty basic.</p>
<p>Here it goes:</p>
<blockquote>
<p>Is it always true that <span class="math-container">$$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$$</span>
where the <span class="math-container">$s_j$</span>'s are positive integers, and may be odd or even?</p>
</blockquote>
<p>We can pe... | Carl Schildkraut | 253,966 | <p>Each factor on the left hand side may be either <span class="math-container">$\equiv 1$</span> or <span class="math-container">$\equiv 3\bmod 4$</span>. If there are an odd number of the latter (i.e. an odd number of odd <span class="math-container">$s_j$</span>), the product will be <span class="math-container">$\e... |
2,055,621 | <p>Prove or disprove the following proposition: </p>
<blockquote>
<p>There are no positive integers $x$ and $y$ such that:
$x^2−3xy+2y^2=10$</p>
</blockquote>
<p>Thanks!</p>
| Erik M | 42,176 | <p>Notice that $x^2−3xy+2y^2=(x-2y)(x-y)=10$.</p>
|
2,055,621 | <p>Prove or disprove the following proposition: </p>
<blockquote>
<p>There are no positive integers $x$ and $y$ such that:
$x^2−3xy+2y^2=10$</p>
</blockquote>
<p>Thanks!</p>
| Behrouz Maleki | 343,616 | <p>$$(x-2y)(x-y)=10=2\times 5=5\times 2=-2\times (-5)=-5\times (-2)$$
or
$$(x-2y)(x-y)=10=1\times 10=10\times 1=-1\times (-10)=-10\times (-1)$$
For example
\begin{cases}
x-2y=2\\
x-y=5
\end{cases}
we have $x=8$ and $y=3$</p>
|
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