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 |
|---|---|---|---|---|
1,071,866 | <p>I am confused by the statement of Sylow's Fourth Theorem:<br>
Let $G$ be a finite group, $p$ a prime. The Sylow $p$-subgroups of $G$ form a single conjugacy class of subgroups.<br>
In particular, I do not understand what it means for the subgroups to form a single conjugacy class?
Thanks!</p>
| Kez | 201,782 | <p>Given a group $G$, you can define an equivalence relation $\sim$ on the set of all subgroups of $G$, where if $H$ and $K$ are subgroups of $G$,
$$H\sim K\iff \text{there exists some }g\in G\text{ such that }K=gHg^{-1}$$
In other words, $H\sim K$ if and only if $K$ is a conjugate of $H$.</p>
<p>Given a subgroup $H$... |
2,273,506 | <p>I was able to simplify a boolean expression $$(\neg a*\neg b*c)+(a*\neg b*\neg c)+(a*b*\neg c)+(a*b*c)$$into the form $$\neg b*(a\oplus c)+a*b$$ where $*$ is the logical and, $+$ is the logical or, and $\oplus$ is the logical XOR.</p>
<p>Apparently, from Wolfram Alpha, this expression can be simplified to $$\left(a... | Fabio Somenzi | 123,852 | <p>I'm not privy to the internals of Wolfram Alpha, but here's a way to derive the expression it produces.</p>
<p>First of all, it's not unreasonable to assume that if the input contains an exclusive OR, WA may decide that the user wants to see exclusive ORs in the output, or is at least OK with them.</p>
<p>Let $f =... |
2,984,918 | <p>How can I prove this? </p>
<blockquote>
<p>Prove that for any two positive integers <span class="math-container">$a,b$</span> there are two positive integers <span class="math-container">$x,y$</span> satisfying the following equation:
<span class="math-container">$$\binom{x+y}{2}=ax+by$$</span></p>
</blockquote... | G.Kós | 141,614 | <p>Consider the equation <span class="math-container">$f(x,y)=x^2+2xy+y^2-(2a+1)x-(2b+1)y=0$</span>.
If <span class="math-container">$a=b$</span> then this is satisfied along the parallel lines <span class="math-container">$x+y=0$</span> and <span class="math-container">$x=y+2a+1$</span>, so we can choose say <span cla... |
4,498,296 | <p>Is there any subtle way to compute the following integral?</p>
<p><span class="math-container">$$\int \frac{\sqrt{u^2+1}}{u^2-1}~ \mathrm{d}u$$</span></p>
<p>The solution i had in mind was substituting <span class="math-container">$u=\tan (\theta)$</span>,then after a few calculations the integral became <span class... | Lai | 732,917 | <p>Letting <span class="math-container">$u=\tan \theta$</span> transforms the integral into
<span class="math-container">$$
\begin{aligned}
\int \frac{\sqrt{u^{2}+1}}{u^{2}-1} d u&=\int \frac{\sec \theta \sec ^{2} \theta d \theta}{\tan ^{2} \theta-1} \\
&=\int \frac{d\theta}{\cos \theta-2 \cos ^{3} \theta} \\
&... |
1,566,215 | <p>Can someone explain to me the difference between joint probability distribution and conditional probability distribution?</p>
| Jay Kim | 564,425 | <p>How can you apply the first equation in the above, i.e. $P(F \, \text{and} \, L)=P(F)*P(L)$? $P(F)=12/25$, $P(L)=7/20$, then the product is $28/125$ which is different from $23/100$ ? What mistake did I make here?</p>
|
1,548,159 | <p>This is a question asked in India's CAT exam: <a href="http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html" rel="nofollow noreferrer">http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html</a> </p>
<blockquote>
<p>How many numbers with distinct digits are possible pr... | kccu | 255,727 | <p>They arrived at $8$ by counting $2$ possible two-digit numbers (47 and 74) plus $3!=6$ possible three-digit numbers (147, 174, 417, 471, 714, 741). $3!$ counts the number of ways to arrange the three digits $1, 4, 7$ that multiply to $28$.</p>
<p>And you <em>can</em> have three digits, as was demonstrated with 147,... |
1,503,958 | <p>In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?</p>
<p>I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!</p>
| Jack D'Aurizio | 44,121 | <p>Set $x=z^2+1$. Then:
$$ \sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+3-8\sqrt{x-1}} = \sqrt{(z-2)^2}+\sqrt{(z-3)^2} = |z-2|+|z-3| $$
equals one for every $z\in[2,3]$, hence for every $x\in[5,10]$.</p>
|
211,903 | <p>Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines $s_1,\dots,s_n$ in $\Bbb R^d$ such that $r_i$ intersects $s_j$ iff there are some permutations $\Pi_1,\Pi_2$ such that $\widetil... | Joseph O'Rourke | 6,094 | <p>As Will Sawin points out, this represents only a partial answer.</p>
<p>Theorem 5 of this paper</p>
<blockquote>
<p>Laison, Joshua D., and Yulan Qing. "Subspace intersection graphs." <em>Discrete Mathematics</em> 310, no. 23 (2010): 3413-3416.
<a href="http://www.sciencedirect.com/science/article/pii/S0012365X... |
211,903 | <p>Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines $s_1,\dots,s_n$ in $\Bbb R^d$ such that $r_i$ intersects $s_j$ iff there are some permutations $\Pi_1,\Pi_2$ such that $\widetil... | Noam D. Elkies | 14,830 | <p>It seems that the matrix
$$
A = \left(\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 & 0 & 1 \\
1 & 1 & 1 &a... |
3,847,209 | <p>I'm currently trying to solve the question below.</p>
<h2>Abed is sitting in front of a large screen tv. He thinks he gets the best view when the screen takes up the maximum angle in his field of view.
What is the optimal point where he sits(to get the largest angle in his view)?</h2>
<p>I think it has something to ... | Narasimham | 95,860 | <p>Assuming the TV surface to be a flat rectangle, the normal to the screen at the center point of the rectangle maximizes planar and solid angles subtended by the viewing eye.</p>
<p>This is a consequence of symmetry of the field being viewed.</p>
<p>Answer after your EDIT:</p>
<p>Assuming the TV to be the red horizon... |
1,179,195 | <p>Good day everyone. </p>
<p>I need to know automata theory. Can you advice me the best way to study math?
What themes will I need to know to understand automata theory. What a sequence of study? What level will I need to study intermediate themes? Maybe can you say something yet, what can help me quickly learn autom... | Alan | 175,602 | <p>Yes, a contraction has a unique fixed point. Proof: Let $d(\cdot,\cdot)$ be our distance function (which is absolute value on the real numbers, but works in more general settings).</p>
<p>Assume $\phi$ is a contraction and $a,b$ are both fixed points. Let $0<k<1$ be our contraction constant. Then $d(g(a),g... |
1,349,654 | <p>Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$</p>
<p>Note :By mathematica,the result is :
$\frac{Gamma\left(\frac1 4\right)Gamma\left(\frac5 4\right)}{\sqrt{\pi}}-\sqrt{2} Hypergeometric2F1\left(\frac1 4,\frac3 4,\frac5 4,\frac1 4\right).$
and ... | Marco Cantarini | 171,547 | <p>We have $$\int_{\pi/4}^{\pi/3}\frac{\sqrt{\tan\left(x\right)}}{\sin\left(x\right)}dx=\int_{\pi/4}^{\pi/3}\frac{1}{\sqrt{\cos\left(x\right)\sin\left(x\right)}}dx=\sqrt{2}\int_{\pi/4}^{\pi/3}\frac{1}{\sqrt{\sin\left(2x\right)}}dx$$ $$=\frac{1}{\sqrt{2}}\int_{\pi/2}^{2\pi/3}\frac{1}{\sqrt{\sin\left(t\right)}}dt
$$ and... |
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| mickep | 97,236 | <p>One way is to use <a href="https://en.wikipedia.org/wiki/Simpson's_rule" rel="nofollow noreferrer">Simpson's rule</a>.</p>
<p>Without it, one could argue what is faster, but: </p>
<p>If $A=(a+b)/2$ and $D=(b-a)/2$, then your first integrand is
$$
(x-A+D)(x-A-D)=(x-A)^2-D^2.
$$
Thus
$$
\begin{aligned}
\int_a^b ... |
185,867 | <p>I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).</p>
<p>So, I'm just curious why don't expert mathematicians use ZF+GCH
instead of ZF+AC(ZFC).</p>
| Asaf Karagila | 7,206 | <p>In some sense $\sf GCH$ is a limiting axiom. While it solves a lot of things, it also means that certain things we are interested in become false or trivialized. And that's no fun.</p>
<p>For example, forcing axioms like $\sf MA$ become trivial assuming even just $\sf CH$, and stronger forcing axioms like $\sf PFA,... |
3,410,850 | <p>The limit law <span class="math-container">$\lim_{x\to c}[f(x)+g(x)]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)$</span> is true given that <span class="math-container">$\lim_{x\to c}f(x)$</span> and <span class="math-container">$\lim_{x\to c}g(x)$</span> both exist. I also know that <span class="math-container">$\lim_{x\to... | gandalf61 | 424,513 | <p>The “area below the curve” interpretation of an integral only makes intuitive sense for functions of one or two variables.</p>
<p>If you have a function of a single variable <span class="math-container">$f(x)$</span> then you can plot the curve <span class="math-container">$y=f(x)$</span> and think of the integral ... |
3,410,850 | <p>The limit law <span class="math-container">$\lim_{x\to c}[f(x)+g(x)]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)$</span> is true given that <span class="math-container">$\lim_{x\to c}f(x)$</span> and <span class="math-container">$\lim_{x\to c}g(x)$</span> both exist. I also know that <span class="math-container">$\lim_{x\to... | Peter Melech | 264,821 | <p>As gandalf61 correctly says You must think of a fourth dimension and the integral could be interpreted as the (signed) area of the "curtain" ( as Khan says) along the curve <span class="math-container">$C$</span> with the height given by values of <span class="math-container">$w=f(x,y,z)$</span> "above" this curve.
... |
2,246,137 | <p>Let
$$\parallel\overrightarrow{a}\parallel =6\text{ and}\parallel\overrightarrow{b}\parallel =3$$</p>
<p>$$2\overrightarrow{a}+(k-3)\overrightarrow{b}\text{ and } k\overrightarrow{a}-\overrightarrow{b}\text{ are parallel}$$</p>
<p>Find all the value(s) of k.</p>
<p>How to get the value(s) of k?</p>
<p>I tried th... | Dr. Sonnhard Graubner | 175,066 | <p>since the vectors $$\vec{2a}+(k-3)\vec{b}$$ and $$k\vec{a}-\vec{b}$$ are parallel
then exists a real number $t$ with
$$2\vec{a}+(k-3)\vec{b}=t(k\vec{a}-\vec{b})$$ this gives
$$\vec {a}(2-kt)+\vec{b}(k-3+t)=\vec{0}$$
from here we get the System
$$2-kt=0$$
$$k-3+t=0$$
with $$t=\frac{2}{k}$$
plugging this in the second... |
2,262,661 | <p>The question is in how many ways can we select 20 different items from the empty set?</p>
<h3>ans:</h3>
<p>Obviously in 0 ways since the empty set has no items. I mean, this seem obvious, but maybe there is a trick to this question.</p>
| Jair Taylor | 28,545 | <p>The answer is pretty clear if you formulate this question slightly more precisely:</p>
<p>How many sets are there of size $20$ are there that are subsets of the empty set?</p>
<p>We have $\{A| A \subseteq \emptyset \text{ and } |A| = 20\} = \emptyset$ since $A \subseteq \emptyset \implies A = \emptyset \implies |A... |
478,713 | <p>I have this logic statement:</p>
<pre><code> (A and x) or (B and y) or (not (A and B) and z)
</code></pre>
<p>The problem is that accessing A and B are rather expensive. Therefore I'd like to access them only once each. I can do this with an if-then-else construct:</p>
<pre><code>if A then
if x then
tru... | Amzoti | 38,839 | <p>Lets graph it and see what happens.</p>
<p><img src="https://i.stack.imgur.com/ZU7lq.png" alt="enter image description here"></p>
<p>Do you see why $f(x) = 0$ at $x=3$ now?</p>
|
478,713 | <p>I have this logic statement:</p>
<pre><code> (A and x) or (B and y) or (not (A and B) and z)
</code></pre>
<p>The problem is that accessing A and B are rather expensive. Therefore I'd like to access them only once each. I can do this with an if-then-else construct:</p>
<pre><code>if A then
if x then
tru... | kiss my armpit | 26,975 | <p>The $x$ intercepts are the abscissa of intersection points of the curve with the $x$ axis. When the curve intersects the $x$ axis, the ordinate is always equal to 0. </p>
<p><img src="https://i.stack.imgur.com/N9Iyb.png" alt="enter image description here"> </p>
<p>The code to generate the graph can be found <a hre... |
1,953,251 | <blockquote>
<p>For what x does the exponential series $P_c(x) = \sum^\infty_{n=0} (-1)^{n+1}\cdot n\cdot x^n$ converge?</p>
</blockquote>
<p><strong>What I got so far:</strong></p>
<p>$\sum^\infty_{n=0} (-1)^{n+1}\cdot n\cdot x^n = (-1)\sum^\infty_{n=0} (-1)^{n}\cdot (\sqrt[n]n)^n\cdot x^n = (-1)\sum^\infty_{n=0}... | Claude Leibovici | 82,404 | <p>May be, you could consider that $$\sum^\infty_{n=0} (-1)^{n+1} n x^n=x\sum^\infty_{n=0} (-1)^{n+1} n x^{n-1}=x \frac{d}{dx} \left(\sum^\infty_{n=0} (-1)^{n+1} x^n \right)=-x\frac{d}{dx} \left(\sum^\infty_{n=0} (-x)^{n} \right)$$ Now, in brackets, you have a simple geometric sum easy to compute (and analyze, for sure... |
1,528,501 | <p>Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ holds for large $n$" or "$P(n)$ holds eventually" and so on.</p>
<p>I wonder if there is in literature a like abbreviat... | Henno Brandsma | 4,280 | <p>In descriptive set theory and logic one sometimes uses $\exists^\ast_n P(n)$ for "there are infinitely many $n$ such that $P(n)$ holds", and $\forall^\ast_n P(n)$ for "all but finitely many $n$ satisfy $P(n)$".</p>
<p>Then at least we have $\lnot \forall^\ast_n P(n) \leftrightarrow \exists^\ast_n \lnot P(n)$, like ... |
1,528,501 | <p>Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ holds for large $n$" or "$P(n)$ holds eventually" and so on.</p>
<p>I wonder if there is in literature a like abbreviat... | yo' | 43,247 | <p>I'm aware of only one way how to write this clearly -- in symbols:</p>
<p>$$ \exists_\infty n\in\mathbb{N} : P(n). $$</p>
<p>However, unless you need this a lot you shouldn't use this notation. If you need it a lot, you can introduce it:</p>
<blockquote>
<p>We suppose that $P(n)$ holds for infinitely many $n$, ... |
2,109,197 | <p><strong>Update:</strong><br>
(because of the length of the question, I put an update at the top)<br>
I appreciate recommendations regarding the alternative proofs. However, the main emphasis of my question is about the correctness of the reasoning in the 8th case of the provided proof (with a diagram).</p>
<p><stro... | pjs36 | 120,540 | <p>Here is another approach that relies on the idea that its easy to detect if three complex numbers lie on a line through the origin, and we attempt to reduce the original problem to this simpler one. </p>
<p>To this end, we might hope that translating our complex numbers (in this case, subtracting $z_1$ from each) d... |
3,639,192 | <p>In their article on the <a href="https://en.wikipedia.org/wiki/Brauer_group#Galois_cohomology" rel="nofollow noreferrer">Brauer group</a> Wikipedia writes:</p>
<blockquote>
<p>Since all central simple algebras over a field <span class="math-container">$K$</span> become isomorphic to the matrix algebra over a sepa... | fleablood | 280,126 | <p>If <span class="math-container">$a^2 + ab = b^2$</span> then <span class="math-container">$a^2 + ab -b^2 =0$</span> and you can solve for <span class="math-container">$a$</span> in terms of <span class="math-container">$b$</span> or <span class="math-container">$b$</span> in terms of <span class="math-container">$a$... |
1,247,185 | <p>I already know, and so ask NOT about, the proof of: <a href="https://math.stackexchange.com/a/463407/53259">$A$ only if $B$ = $A \Longrightarrow B$</a>.<br>
Because I ask only for intuition, please do NOT prove this or use truth tables. </p>
<p><strong>My problem:</strong> I try to avoid memorisation. So wh... | hmakholm left over Monica | 14,366 | <p>I understand "A only if B" as "A can be true <em>only</em> in those possible worlds where B is true also".</p>
|
4,116,134 | <p>Find angle between <span class="math-container">$y=\sin x$</span> and <span class="math-container">$y=\cos x$</span> at their intersection point.</p>
<p>Intersection points are <span class="math-container">$\frac{\pi}{4}+\pi k$</span> and to find angle between them we need to compute derivatives at intersection poin... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>If <span class="math-container">$u$</span> is the angle <span class="math-container">$$\tan u=\left|\dfrac{\cos x-(-\sin x)}{1+\cos x(-\sin x)}\right|$$</span></p>
<p>At the point of intersection,</p>
<p><span class="math-container">$$\cos x+\sin x=\sqrt2\sin(x+\pi/4)=\cdots=\sqrt2(-1)^n$$</span></p>
<p... |
866,808 | <p>In my lecture notes:</p>
<p>Let $m,n\in \mathbb{N}$ be relatively prime. The fundamental theorem of arithmetic implies that each divisor of $mn$ is the product of two unique positive relatively prime integers $d_1|m$ and $d_2|n$.</p>
<p>Please could someone help me understand how this is implied? I have no idea</p... | TonyK | 1,508 | <p>$d_1$ and $d_2$ are not, in general, prime numbers. They are <em>relatively</em> prime (to each other); that means that the highest common factor of $d_1$ and $d_2$ is $1$.</p>
|
4,151,381 | <p>I am a nube just getting into mathematics and set theory.</p>
<p>I am learning about how we can produce the list of ordinal numbers by purely using the null set, with 0 standing for Ø, 1 standing for {Ø}, 2 standing for {Ø, {Ø}} and so forth. What I am confused about is the operation at play here to produce the larg... | HFKy | 534,596 | <p>By contradiction, suppose <span class="math-container">$h(\Omega(f))\subsetneq \Omega(g)$</span>. We can assume <span class="math-container">$Y = \Omega(g)$</span> since <span class="math-container">$K = h^{-1}(\Omega(g))$</span> is compact and <span class="math-container">$f-$</span>invariant, so we have <span clas... |
4,277,924 | <p><strong>1. p ∧ ¬q = T</strong>
<br><br>
<strong>2. (q ∧ p) → r = T</strong>
<br><br>
<strong>3.¬p → ¬r = T</strong>
<br><br>
<strong>4.(¬q ∧ p) → r = T</strong>
<br><br>
From <strong>Eq 1</strong>, we got <strong>p = T</strong> and <strong>q = F</strong>
<br>
Now Apply value of <strong>P</strong> in <strong>Eq 3</st... | mohottnad | 955,538 | <p>It's perfectly ok that your set of sentences is consistent if you have 2 different models satisfying your set of sentences (a theory) since consistency has nothing to do with uniqueness of model as referenced <a href="https://en.wikipedia.org/wiki/Consistency" rel="nofollow noreferrer">here</a></p>
<blockquote>
<p>a... |
4,277,924 | <p><strong>1. p ∧ ¬q = T</strong>
<br><br>
<strong>2. (q ∧ p) → r = T</strong>
<br><br>
<strong>3.¬p → ¬r = T</strong>
<br><br>
<strong>4.(¬q ∧ p) → r = T</strong>
<br><br>
From <strong>Eq 1</strong>, we got <strong>p = T</strong> and <strong>q = F</strong>
<br>
Now Apply value of <strong>P</strong> in <strong>Eq 3</st... | ryang | 21,813 | <p>A set of sentences as consistent iff their conjunction is satisfiable.</p>
<p>(Informally: a consistent system is one whose premises/axioms are coherent in <em>some</em> universe.)</p>
<p>So, in propositional logic, an inconsistent system is one whose conjunction is a contradiction, i.e., whose conjunction is false ... |
4,238,241 | <blockquote>
<p>Let tangents <span class="math-container">$PA$</span> and <span class="math-container">$PB$</span> on hyperbola from any point <span class="math-container">$P$</span> on the Director Circle of hyperbola such that <span class="math-container">$d(P,AB).d(C,AB)=4d(S_1,PA).d(S_2,PA)$</span> and <span class=... | Math Lover | 801,574 | <p>It is not a horizontal or vertical hyperbola - it has one branch which is entirely in the first quadrant and with other information provided, this fact will become clear as we go through the solution steps.</p>
<p>There are properties of hyperbola that would hold regardless of whether you rotated or shifted the coor... |
124,060 | <p>I have a small dark object in the upper left corner of an <code>image</code>.</p>
<p>How can I separate it from the noisy rest and determine its <code>IntensityCentroid</code>?</p>
<p><a href="https://i.stack.imgur.com/D7pHv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/D7pHv.png" alt="enter ima... | Svyatoslav Korneev | 42,293 | <p>Let me give you my general solution for the scientific image processing. The most important is to try to use less subjective parameters, like a threshold value. First, let's assume that the image is taken from charge-coupled device (CCD) or CMOS matrix. These detectors are widely used in most of the modern digital e... |
1,509,007 | <blockquote>
<p>Consider an unweighted and undirected graph $G=(V,E)$, where the vertices $V$ of $G$ lie on the unit n-sphere. If we choose a normal vector uniformly at random on this $n$-sphere, then the corresponding hyperplane (shifted to go through the origin) splits the vertices into two disjoint sets $A$ and $... | Tenno | 286,106 | <p>It isn't clear to me exactly what you are asking the community to provide, but you can compare this to previously established bounds to get a sense for how well you are doing. In particular, a mathematician named Robin showed that
$$
\sigma(n) < e^{\gamma}n\log(\log(n)) + \frac{.65n}{\log(\log(n))}
$$
holds for a... |
1,030,274 | <p>$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?</p>
| Andrea Mori | 688 | <p>Consider the ideal $I=(2,X)$ in $\Bbb Z[X]$.</p>
<p>Concretely, it consists of polynomials with integer coefficients of the form $a_0+a_1X+\cdots$ with $a_0$ even.</p>
<p>It cannot be generated by a single element $P(x)$ because you can never get $2$ and $X$ both multiples of a single polynomial.</p>
<p>But it is... |
267,051 | <p>Games appear in pure mathematics, for example, <a href="https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game" rel="noreferrer">Ehrenfeucht–Fraïssé game</a> (in mathematical logic) and <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game" rel="noreferrer">Banach–Mazur game</a> (in topo... | Eilon | 64,609 | <p>You can read about the application of Blackwell's approachability theory, a nontrivial area in game theory, to normal numbers in <a href="https://www.jstor.org/stable/30035680?seq=1#page_scan_tab_contents" rel="nofollow noreferrer">https://www.jstor.org/stable/30035680?seq=1#page_scan_tab_contents</a></p>
|
1,833,854 | <p>We have $M$ Binomial random variables, where $X_0 \sim $ Bin$(n,p)$ and $X_i \sim $ Bin$(n,1/2)$. </p>
<p>Suppose $p > 1/2$. I'm interested in the probability that $\mathbb{P}(\max \{X_1,\dots,X_M\} \geq X_0)$. Is this tractable? </p>
<p>If not, is it tightly boundable/approximable? If this is a very difficult ... | Jason | 164,082 | <p>$P( \max \{X_1,...,X_M \} \geq X_0)=P(X_i \geq X_0$ for some i) = $1-P( \text{each }X_i < X_0)$.
<br>
Now $P( \text{each }X_i < X_0)=\sum_{k=0}^n P(\text{each }X_i<X_0 \vert X_0=k) \cdot P(X_0 =k)$ =$\sum_{k=0}^n P( \text{each } X_i<k \text{ and } X_0=k)$ = $\sum_{k=1}^n \big[ P(X_1<k) \big]^M \cdot P... |
2,748,495 | <p>I have a numeric table for artillery operations (Royal Italian Army, year 1940), in the instructions it refers to a measure of a planar angle as $32.00^{\circ\circ}$ and it seems to me that this angle is equivalent to $\pi$ radian. I had a look at <a href="https://en.wikipedia.org/wiki/Gradian" rel="nofollow norefer... | Christian Blatter | 1,303 | <p>The full circle is subdivided into $6400$ <em>artillery promilles</em>, whereby "pro mille" is latin for ${1\over1000}$. The idea behind this angle measure is that a circle of radius $1000$ m has a circumference of $$2\pi\cdot 1000\ {\rm m}=6283.2\ {\rm m}\ \approx 6400\ {\rm m}\ ,$$ so that an angle of $1$ artiller... |
113,349 | <p>I have a lot of old <em>Mathematica</em> version 2.2 <code>.ma</code> files that simply crash the current (10.4.1) version of <em>Mathematica</em> if I try to open them.</p>
<p>Is there some way to convert them to <code>.nb</code> notebook files?</p>
<p>Is 5.2 the latest version that will do the conversion? And if... | Alexey Popkov | 280 | <p><a href="http://forums.wolfram.com/mathgroup/archive/2012/May/msg00371.html" rel="nofollow">This</a> <a href="https://groups.google.com/forum/#!forum/comp.soft-sys.math.mathematica" rel="nofollow">MathGroup</a> discussion should answer your question, so I'll cite it here:</p>
<blockquote>
<p>On Tue, 29 May 2012 0... |
52,802 | <p>This is partly a programming and partly a combinatorics question.</p>
<p>I'm working in a language that unfortunately doesn't support array structures. I've run into a problem where I need to sort my variables in increasing order.</p>
<p>Since the language has functions for the minimum and maximum of two inputs (b... | James | 68,019 | <p>Following Sleziak's suggestion, it might be useful to point out that under the bijection $A\mapsto\chi_{A}$, the symmetric difference is realized as </p>
<p>\[ A\triangle B\sim\left|\chi_{A}-\chi_{B}\right|^{2}.\]</p>
<p>It is then straightforward to check that multiplication is distributive: </p>
<p>\[ C\cap\lef... |
2,439,744 | <p>How to prove that $\lim_{x\to 0,y\to 0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$?</p>
<p>I already calculated that when a = 1 by multiplying $\sqrt {a+x^2y^2} + 1$ on both denominator and numerator and using the fact $x^2y^2<(x^2+y^2)^2/4$.</p>
<p>Any help will be appreciat... | Parcly Taxel | 357,390 | <p>Evaluating on the path $x=y$:
$$\lim_{x,y\to0}{\frac{\sqrt{a+x^2y^2}-1}{x^2+y^2}}=\lim_{x\to0}{\frac{\sqrt{a+x^4}-1}{2x^2}}=\lim_{x\to0}{\frac{\sqrt a-1}{2x^2}}$$
If $a\ne1$ then $\sqrt a-1\ne0$ and a singularity exists at $x=y=0$, so the limit does not exist.</p>
|
2,662,605 | <p>Problem: If ${F_n}$ is a sequence of bounded functions from a set $D \subset \mathbb R^p$ into $ \mathbb R^q$ and if ${F_n}$ converges uniformly to $F$ on $D$, then $F$ is also bounded. </p>
<p>Proof(Attempt): Let $\epsilon >0$. Since ${F_n}$ converges uniformly to $F$ on $D$, then there is an $N \in\mathbb R$ s... | spaceisdarkgreen | 397,125 | <p><strong>HINT</strong></p>
<p>We have $$ \| F(x) \| = \|F(x)-F_n(x) + F_n(x)\| \le \|F(x)-F_n(x)\| + \|F_n(x)\|.$$</p>
|
251,559 | <p>First of all, this is a very silly question, but I finished high school a long time ago and I really don't remember much about some basic stuff.</p>
<p>I have:</p>
<p>$$T(x) = -\frac1{10}x^2 + \frac{24}{10}x - \frac{44}{10}$$</p>
<p>$T$ = temperature</p>
<p>$x$ = hour</p>
<p>And they ask me to find the hour in ... | Jebruho | 40,030 | <p>Use the quadratic formula so that the zeroes of the function occur at $\frac{-b \pm \sqrt {b^{2}-4ac}}{2}$, where $a=\frac{-1}{10},b=\frac{24}{10}x,c=\frac{-44}{10}$. In general, polynomials of up to degree 4 have solutions in roots, but solutions of degree 5 or higher do not.</p>
|
251,559 | <p>First of all, this is a very silly question, but I finished high school a long time ago and I really don't remember much about some basic stuff.</p>
<p>I have:</p>
<p>$$T(x) = -\frac1{10}x^2 + \frac{24}{10}x - \frac{44}{10}$$</p>
<p>$T$ = temperature</p>
<p>$x$ = hour</p>
<p>And they ask me to find the hour in ... | Thomas | 26,188 | <p>You want to solve the quadratic equation:</p>
<p>$$
0 = -\frac{1}{10}x^2 + \frac{24}{10}x - \frac{44}{10}
$$
You can indeed just apply the quadratic formula, but first you can make things a tiny bit easier by multiplying by -10 on both sides to get
$$
0 = x^2 - 24x + 44.
$$
Now if $a$ and $b$ are the solutions you... |
4,198,496 | <p>If the value of <span class="math-container">$\int_{1}^2{e^{x^{2}}}dx$</span> is <span class="math-container">$\alpha$</span>, then what is <span class="math-container">$$\int_{e}^{e^{4}}{\sqrt{\log x}}dx$$</span></p>
<p>It just seems that substitution of any sort does not help. Is there some other way in which thes... | Tito Eliatron | 84,972 | <p>Take <span class="math-container">$t=\sqrt{\log x}$</span> then <span class="math-container">$x=e^{t^2}$</span> and <span class="math-container">$dx=2te^{t^2}$</span>. So
<span class="math-container">$$I=\int_e^{e^4}\sqrt{\log x}dx=\int_1^22t^2e^{t^2}dt.
$$</span>
Using integration by parts: <span class="math-contai... |
1,225,359 | <p>Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is:</p>
<p>What I tried to do was make pairs of two terms and the expand it as a binomial expression and then again expand the binomial in the resulting series which gave me an expression with lot of un... | lab bhattacharjee | 33,337 | <p>$$(x+y+z+w)^6=\{y+(x+w+z)\}^6=\cdots+(x+w+z)^6$$</p>
<p>$$(x+w+z)^6=\sum_{r=0}^6\binom6rx^{6-r}(w+z)^r$$</p>
<p>We need $r\ne6$</p>
<p>The sum of the reuqired coefficients should be $\sum_{r=0}^5\binom6r(1+1)^r$ (setting $x=w=z=1$)</p>
<p>$$=\sum_{r=0}^6\binom6r2^r-\binom662^6=(1+2)^6-2^6$$</p>
|
3,146,161 | <blockquote>
<p>Find particular solution of <span class="math-container">$\dfrac{dx}{dy} +x\cot y =y\cot y$</span> given <span class="math-container">$x=0$</span> when <span class="math-container">$y= π/2$</span>.</p>
</blockquote>
<p>Please help me figure this out
I can't separate them out in order to integrate</p... | Henry Lee | 541,220 | <p>For an integral to be convergent:
<span class="math-container">$$\int_0^\infty f(x)dx=I$$</span>
where <span class="math-container">$I$</span> is a real number. We need the function to decrease such that:
<span class="math-container">$$\lim_{x\to\infty}f(x)=0$$</span>
however for this function,
<span class="math-con... |
2,482,868 | <p>I am trying to find</p>
<p>$$ \lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}],$$</p>
<p>where $p>0$. I have tried to factor out as</p>
<p>$$(1+x^{p+1})^{\frac1{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} =x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... | user236182 | 236,182 | <p>The question says $p>0$. Let $x=\frac{1}{t}$. $$\lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}]=$$</p>
<p>$$=\lim_{t\to 0^+}\frac{(t^{p+1}+1)^{\frac{1}{p+1}}-1}{t}-\lim_{t\to 0^+}\frac{(t^p+1)^{\frac{1}{p}}-1}{t}=$$</p>
<p>$$=((t^{p+1}+1)^{\frac{1}{p+1}})'|_{t=0}-((t^p+1)^{\frac{1}{p}})... |
685,642 | <p>I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.</p>
<p>I don't understand though how to generalize for the entries on th... | user44197 | 117,158 | <p>Note that trace equals zero says that the $n,n$ term is given by the remaining. So start with the usual basis with
$$
A_{k,m}=1, ~~~ (k,m) \ne (n,n),\\ A_{i, j} = 0 , ~~~~(i,j) \ne (k, m), \\
A_{n,n} = -\sum_{i=1}^{n-1} A_{i,i}$$</p>
<p>Thus the basis has $n^2-1$ elements.</p>
|
1,437,073 | <p>Ok guys, I have to solve this ODE</p>
<p>$$
\frac{d^2y}{dx^2}=f(x), \quad
x>0,\quad y\left(0\right) = 0, \quad
\left.\frac{dy}{dx}\right\lvert_{x=0}=0
$$
The solution I should get is in the form of
$$y\left(x\right)=\int_0^x k\left(t\right)\, dt $$
Moreover, I should tell what the function $\,k\left(t\right... | obataku | 54,050 | <p>The Green's function for $\dfrac{d^2}{dx^2}$ is the ramp function $x\theta(x)$, where $\theta(x)$ is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" rel="nofollow">Heaviside step function</a>. We can write the solution for the general inhomogeneous problem as follows. $$\begin{align*}y(x)&=\in... |
509,928 | <p>There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake.</p>
<p>I want to divide the cake into 2 non-empty rectangular pieces, by either a horizontal or a vertical cut, such that the nu... | Jeff Snider | 119,951 | <p>Supposing the cake is open, and the toppings are open, we can position them so that a cut between edge and topping is impossible, and cuts between adjacent toppings are possible.</p>
<p>It appears that $\lfloor \frac{N}{4} \rfloor$ is the maximal case for a given $N>1$ on your two-dimensional cake, as demonstrat... |
4,341,297 | <p>Evaluate the given expression <span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}$$</span> The given answer is <span class="math-container">$\dfrac{1}{4}$</span>. My attempt:
<span class="math-container">$$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^{2n}\cdot2}}\\=... | Mastrem | 253,433 | <p>Using the identity <span class="math-container">$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$</span>, we find that
<span class="math-container">$$
\begin{align*}
\sqrt[n]{\frac{20}{2^{2n+4}+2^{2n+2}}} &= \sqrt[n]{\frac1{2^n}}\cdot\sqrt[n]{\frac{20}{2^4+2^2}}\\
&= \sqrt[n]{\frac1{4^n}}\cdot\sqrt[n]{\frac{20}{16+4}}\\... |
2,685,424 | <p>What I don't understand is that why can't we find the general solution of non homogeneous differential equation from the non homogeneous one itself. Currently we use the homogeneous equation also. </p>
<p>Why isn't it that general solution is not available from the non-homogeneous equation itself?</p>
| symplectomorphic | 23,611 | <p>The whole point is that if $x_1$ and $x_2$ solve $L(x)=y$, where $L$ is a linear differential operator, then $$0=y-y=L(x_1)-L(x_2)=L(x_1-x_2)$$ so $x_1$ and $x_2$ differ by a homogeneous solution. Note that linearity is crucial here.</p>
<p>(You seem to think the inhomogeneous equation has a <em>unique</em> particu... |
501,250 | <p>I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ and $\textbf{y}=(y_1,y_2) \in \text R^2$. If true, I'd also like to extend this to $\textbf{x} \in \text R^{n_1+n_2}$... | Prahlad Vaidyanathan | 89,789 | <p>$C$ is not closed because $0$ is a limit point of $C$, but $O \notin C$.</p>
<p>$C$ is not open because, for any $x = 1/n \in C$, and any $\delta > 0$, there is a point $z\in \mathbb{R}\setminus C$ such that $x-\delta < z < x+\delta$</p>
|
63,348 | <p>This question arises from a discussion with my friends on a commonly encountered IQ test questions: "What's the next number in this series 2,6,12,20,...". Here a "number" usually means an integer. I was wondering whether there is a systematical way to solve such problems.Let us call a point on a plane integer point ... | Robert Israel | 13,650 | <p>Let the points be $(x_j, y_j), j=1\ldots n$. If the $x_j$ are consecutive, the Lagrange interpolating polynomial will take integers to integers: easy proof by induction, using the difference operator $\Delta(p)(x) = p(x+1) - p(x)$. If not, choose arbitrary integers for the $y$ values to fill in the gaps. </p>
|
1,710,517 | <blockquote>
<p>Show that 13 is the largest prime which divides two consecutive terms of $n^2 + 3$.</p>
</blockquote>
<p>The integers are $39$ and $52$. First of all, I set the variable for the number as $k$. So, $k|n^2 +3$ and $k|n^2 + 2n+ 4$ which imply that $k|2n+1$. $n=6$ over here. And the fact that 13 is the l... | paw88789 | 147,810 | <p>If $k\mid n^2+3$ and $k\mid n^2+2n+4$, then as you noted, $k\mid 2n+1$.</p>
<p>But then also from $k\mid n^2+3$ we have $k\mid 2n^2+6$, and from $k\mid 2n+1$ we have $k\mid 2n^2+n$.</p>
<p>Hence $k\mid (2n^2+n)-(2n^2+6))=n-6$. And so $k\mid 2n-12$.</p>
<p>From $k\mid 2n-12$ and $k\mid 2n+1$, we obtain $k\mid 13$.... |
4,292,815 | <p>Compute line integral <span class="math-container">$\int_a^b (y^2z^3dx + 2xyz^3dy + 3xy^2z^2dz)$</span> where <span class="math-container">$a = (1,1,1)$</span> and <span class="math-container">$b = (2,2,2)$</span></p>
<p>What I have done:</p>
<p>To find <span class="math-container">$t$</span> I used the calculation ... | Community | -1 | <p>The <a href="https://math.stackexchange.com/questions/4292815/compute-line-integral-int-ab-y2z3dx-2xyz3dy-3xy2z2dz/4292824#comment8938860_4292815">hint</a> you mentioned implies that you can use the <a href="https://en.wikipedia.org/wiki/Gradient_theorem" rel="nofollow noreferrer">fundamental theorem of calculus</a>... |
858,353 | <p>$$x^2(y')^2+3xyy'+2y^2=0$$
I have no idea how to start, I probably need to do some tricky substitution but as of now I cant see any options.</p>
| Community | -1 | <p>It factors as $$(xy'+2y)(xy'+y)=0.$$ Now solve each equation separately.</p>
|
203,827 | <p>Suppose I have the following lists: </p>
<pre><code>prod = {{"x1", {"a", "b", "c", "d"}}, {"x2", {"e", "f",
"g"}}, {"x3", {"h", "i", "j", "k", "l"}}, {"x4", {"m",
"n"}}, {"x5", {"o", "p", "q", "r"}}}
</code></pre>
<p>and </p>
<pre><code>sub = {{"m", "n"}, {"o", "p", "r", "q"}, {"g", "f", "e"}};
</code><... | kglr | 125 | <p>You can also turn <code>prod</code> into an <code>Association</code> using sorted second elements as keys and then use <a href="https://reference.wolfram.com/language/ref/Lookup.html" rel="nofollow noreferrer"><code>Lookup</code></a>:</p>
<pre><code>Lookup[Sort/@sub] @ GroupBy[ Sort@*Last] @ prod
</code></pre>
<bl... |
162,611 | <p>I am working with the square-roots of square symmetric matrices. The answers are to be binary symmetric matrices.</p>
<p>If we take the matrix $$M = \begin{pmatrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&0&1&0&... | Will Jagy | 10,400 | <p>Gerry Myerson gave a recipe for finding a symmetric solution, also providing a solution in a comment. I wanted to implement that while preserving the structure of the Don Giles matrix for order 2 (Fano plane). That is, I wanted the first four rows and first four columns to be the analogous pattern to the 7 by 7. Ger... |
839,431 | <p>Can someone be kind enough to show me the steps to integrate this, I'm sure it's by parts but how do I split up the sin part?
$$x\sin(1+2x)$$</p>
| user1729 | 10,513 | <p>I would start with an integration by substitution first. So, let $z=1+2x$ and note that this implies $x=\frac{z-1}{2}$, so we obtain the following. $$\frac14\int(z-1)\sin z\operatorname{d}z=\frac14\int z\sin\operatorname{d}z-\frac14\int \sin z\operatorname{d}z$$
Then, integrating $\sin z$ is standard, while $z\sin z... |
253,746 | <p>I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger?</p>
<p>1.)If the region $U$ is simply connected, then for every closed curve contained therein, the integral of the holomorphic function $f$ defined on $U$ over the curve is zero.</p>
<p>2.) If the domain $D$ is... | Julián Aguirre | 4,791 | <p>The second one is stronger, since any closed curve in a simply connected domain satisfies the condition imposed on the second version of Cauchy's theorem.</p>
<p>Somewhat more general versions of the theorem are called the homotopical version (corresponding to your first theorem) and the homological version (corres... |
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | H. Hasson | 2,665 | <p>Hopefully I'll have some time later to ellaborate, but for now - here is a great reference I wish somebody had shown me when I started out, for how to attack questions of this type:
<a href="http://books.google.com/books?id=l0DgAIx_djoC&printsec=frontcover&dq=waterhouse+affine&source=bl&ots=nup7qU4Al... |
497,546 | <p>Let $A$ be an infinite set.</p>
<p>Then, we can construct an injective function $f:\omega \rightarrow A$. </p>
<p>But how do i construct this via orginal statement of $AC_\omega$? (i.e. $\forall countable X, [\emptyset \notin X \Rightarrow \exists f:X\rightarrow \bigcup X \forall A\in X, f(A)\in A$)</p>
<p>So my ... | You_Don't_Know_Who | 484,838 | <p>Hint: Actually, one can simply use binomial theorem to get the expansion of <span class="math-container">$(1-q)^{1/2}$</span>. Then, substitute <span class="math-container">$q=2xt-t^2$</span>. Again, use binomial expansion on the powers of <span class="math-container">$q$</span>. Combining the two expansions, you ca... |
2,759,407 | <p>Suppose there's a exam with 5 questions.
If the probability that Student $1$ correctly answers question $i$ is $P_{1.i}$, then</p>
<p>$P_{1.1} = 0.3$ , $P_{1.2} = 0.4$ , $P_{1.3} = 0.9$, $P_{1.4} = 0.7$ , $P_{1.5} = 0.1$</p>
<p>For Student $2$, </p>
<p>$P_{2.1} = 0.4$ , $P_{2.2} = 0.5$ , $P_{2.3} = 0.2$, $P_{2.4}... | Henno Brandsma | 4,280 | <p>Compute all probabilities $P(C_i = j)$ where $C_i$ is the number of correct questions scored by student $i$. These are just sums of products of the $P_{i,j}$ really. Then sum all $P(C_1 = j)P(C_2 = k)$ over all pairs $j > k$ that are allowed. This is allowed by independence of the scores of both students.</p>
<p... |
2,292,096 | <p>I'm learning inequalities for the first time, and except a paragraph by Paul Zeitz in his book Art and Craft of problem solving, none actually give much motivation of why should I care about inequalities.</p>
<p>The example given by Paul Zeitz was that to prove $b^2-b+1$ is never a perfect square for integer $b$. W... | DanielWainfleet | 254,665 | <p>(I). Many Q's are readily answered by the following: Let $J$ be a (bounded or unbounded) interval of $\mathbb R.$ Let $f:J\to \mathbb R$ with $f''(x)<0$ for all $x\in J.$ Then when $w_1,...,w_n$ are non-negative with $\sum_{i=1}^nw_i=1,$ and $x_1,...,x_n \in J$ then $$(*)\quad f\left(\sum_{i=1}^nw_ix_i\right)\geq... |
4,258,226 | <p>Suppose a group <span class="math-container">$G$</span> splits as a semidirect product <span class="math-container">$N\rtimes\mathbb{Z}_2$</span>, and let <span class="math-container">$\phi:G\to\mathbb{Z}_2$</span> the the associated quotient map. If I have a subset of elements <span class="math-container">$\{g_1,\d... | Derek Holt | 2,820 | <p>A counterexample is the Coxeter Group <span class="math-container">$$G =W(A_3) = \langle x,y,z \mid x^2,y^2,z^2,(xy)^3,(yz)^3,(xz)^2 \rangle.$$</span></p>
<p>Then <span class="math-container">$G$</span> is isomorphic to <span class="math-container">$S_4$</span> with <span class="math-container">$x,y,z \mapsto (1,2),... |
1,950,077 | <blockquote>
<p>A standard deck of cards consists of 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). Suppose you shuffle such a deck and draw three cards at random without replacement. Let <span class="math-container">$A_i =$</span> the event that the <span class="math-container">$i$</span>t... | Ross Millikan | 1,827 | <p>The formal answer is that the first card is red or black with probability $\frac 12$. The probability the second card is red is then $\frac{25}{51} \cdot \frac 12$ (first card is red) + $\frac{26}{51} \cdot \frac 12$ (first card is black)$=\frac 12$, so it is false. The intuitive answer is to draw the two cards. ... |
2,275,679 | <p>Originally, I want to show that
$$
\frac{\sqrt{a \cdot b + \frac{b}{a}x^2}\arctan \left(\frac{c}{\sqrt{a \cdot b + \frac{b}{a}x^2}}\right)}{\sqrt{a \cdot b}\arctan \left(\frac{c}{\sqrt{a \cdot b}}\right)} \geq 1 \ \ \text{for} \ \ x, a,b,c > 0 \ .
$$
To do so, I figured it is sufficient to show that
$$
f(x) = ... | Community | -1 | <p>Here's an explanation that mixes varying amounts of intuition and slight rigor.</p>
<hr>
<p><strong>Volume of a solid of revolution:</strong></p>
<p>When using the disk method the idea is that we're adding up the volumes of a massive amount of extremely thin disks between $x=a$ and $x=b$ in order to get the volum... |
2,904,276 | <p>I have this, what seems basic triangle problem, and not certain if the given information is sufficient to solve the problem of finding an angle.
We have one main isoceles and another one inside of it.
I have attached here a diagram, and we wish to find the angle in red:</p>
<p><a href="https://i.stack.imgur.com/moR... | Donald Splutterwit | 404,247 | <p>Let $\widehat{QRP}= \widehat{RPQ}=x$, so $\widehat{RQP}=180-2x$, so $\widehat{RQS}=156-2x$, so the sum of the angles $\widehat{QST}$ and $ \widehat{QTS}$ is $24+2x$, so they are both $x+12$, so $\widehat{RTS}=168-x$, so $\widehat{RST}=12$.</p>
|
2,904,276 | <p>I have this, what seems basic triangle problem, and not certain if the given information is sufficient to solve the problem of finding an angle.
We have one main isoceles and another one inside of it.
I have attached here a diagram, and we wish to find the angle in red:</p>
<p><a href="https://i.stack.imgur.com/moR... | Edward Porcella | 403,946 | <p>Although the solution of @Donald Splutterwit might seem like algebra and equations, it is really straight-forward geometry, using the principle that the three angles of a triangle sum to $180^o$.</p>
<p>Perhaps the following may be another helpful way to exhibit the geometry of the situation.
<a href="https://i.sta... |
2,041,484 | <p>Solve the system of equations for all real values of $x$ and $y$
$$5x(1 + {\frac {1}{x^2 +y^2}})=12$$
$$5y(1 - {\frac {1}{x^2 +y^2}})=4$$</p>
<p>I know that $0<x<{\frac {12}{5}}$ which is quite obvious from the first equation.<br>
I also know that $y \in \mathbb R$ $\sim${$y:{\frac {-4}{5}}\le y \le {\frac 4... | ILoveMath | 42,344 | <p>Hint: Notice $x,y \neq 0 $</p>
<p>$$ \frac{12}{5x} + \frac{4}{5y} = 2 $$</p>
|
4,019,748 | <p>Let <span class="math-container">$H_1=(H_1, (\cdot, \cdot )_1)$</span> and <span class="math-container">$H_2=(H_2, (\cdot, \cdot )_2)$</span> be Hilbert spaces. Suppose that <span class="math-container">$H_1$</span> is continuously and densely embedded in <span class="math-container">$H_2$</span>. Simbolically, <spa... | JLMF | 551,373 | <p>This doesn't seem true to me without more restrictions.
For example, take <span class="math-container">$X$</span> the zero vector space in <span class="math-container">$H_1$</span>, then <span class="math-container">$X$</span> is not dense in <span class="math-container">$Y$</span>.</p>
<p>Edit: What if we consider ... |
204,043 | <p>I am looking at the following optimization problem</p>
<p><span class="math-container">$$
\begin{align*}
\max\ & 1000 r_1 + \frac{1}{2}r_2 + \frac{1}{3}r_3\\
\text{s.t. }& 1000^2 r_1 + \frac{1}{4}r_2 + \frac{1}{9}r_3 = \frac{1}{9},\\
& 1000^2 p_1 + \frac{1}{4}p_2 + \frac{1}{9}p_3 = \frac{1}{9},\\
& ... | A.G. | 7,060 | <p>It looks like a bug. If you define the <code>error</code> variable like this</p>
<pre><code>sol = FindMaximum[{
1000 r1 + 1/2 r2 + 1/3 r3,
{
1000^2 r1 + 1/4 r2 + 1/9 r3 == 1/9,
r1 + r2 + r3 + r4 == 2,
r1^2 <= p1,
r2^2 <= p2,
r3^2 <= p3,
r4^2 <= p4,
1000^2 p1 + 1/4 p2 + ... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| zyx | 14,120 | <p>$x \to x^p$ is an automorphism sending $r$ to $r-a$ for any root $r$ of the polynomial. This operation is cyclic of order $p$, so that one can get from any root to any other by applying the automorphism several times. The Galois group thus acts transitively on the roots, which is equivalent to irreducibility.</p>
|
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Jyrki Lahtonen | 11,619 | <p>Greg Martin and zyx have given you IMHO very good answers, but they rely on a few basic facts from Galois theory and/or group actions. Here is a more elementary but also a longer approach. </p>
<p>Because we are in a field with $p$ elements, we know that $p$ is the characteristic of our field. Hence, the polynomial... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Utkarsh | 39,872 | <p>$x^p-x+a$ divides $x^{p^p}-x$. If $f$ is an irreducible divisor of $x^p-x+a$ of degree $d$ then $\mathbf{Z}_p[x]/f$ will be a subfield of the field with $p^p$ elements so $p^p = (p^d)^e$ and so $d=1$ or $e=1$. since $x^p-x+a$ has no roots $e=p.$</p>
|
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Community | -1 | <p>The supposition of Greg Martin is truth, if a polinomyal $f$ with $deg(f)=n$ satisfies the property, then $n\ge p$, by contradiction argument, just write the expansion with the Newton's formula and analyse the coeficient of $x^{n-1}$ term, you get $\binom n 1a_{n}+a_{n-1}=a_{n-1}$, if $n\lt p$, this equation is an a... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Community | -1 | <p>Below a suggestion for the 'What about open problems' part of What topics can I ask about here, ie presently the old FAQs. It is based on various contributions and input for details see <a href="https://meta.mathoverflow.net/questions/360/what-should-be-the-policy-on-open-problems-on-mo">What should be the policy on... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Danny Ruberman | 3,460 | <p>Two small things I noticed on "<a href="https://mathoverflow.net/help/on-topic">on-topic</a>": The link <a href="https://meta.mathoverflow.net/questions/882/how-to-write-a-good-mathoverflow-question">How to ask a good MathOverflow question</a> seems to be broken. Also, in the subtopic "MathOverflow is not a discuss... |
369,589 | <p>I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite intriguing. I am considering learning stochastic calculus myself, but do not quite know what kind of prerequisites sh... | Alexander Sokol | 28,924 | <p>Stochastic calculus relies heavily on martingales and measure theory, so you should definitely have a basic knowledge of that before learning stochastic calculus. Basic analysis also figures prominently, both in stochastic calculus itself (where limit procedures of various kinds appear, as well as the occasional Hil... |
2,610,501 | <p>A five digit number has to be formed by using the digits $1,2,3,4$ and $5$ without repetition such that the even digits occupy odd places. Find the sum of all such possible numbers.</p>
<p>This question came in my test where you literally get $2$ minutes to solve one problem. I want to how to solve this problem mor... | Arnaud Mortier | 480,423 | <p>Choose a position for the $2$. How many of these numbers have the $2$ in that position? There are two options for the $4$ and $3!$ options for the remaining $3$ numbers, that gives you $12$ numbers. Hence the $2$ brings $12\times 20202$ to the sum.
Similarly the $4$ brings $12\times 40404$.</p>
<p>Now if you fix an... |
2,485,425 | <p>If $A$ is a non empty subset of the reals and $f$ is a bounded function from $A$ to the reals, how can we show that:</p>
<blockquote>
<p>$\sup|f(x)| - \inf|f(x)| \le \sup(f(x)) - \inf(f(x))$?</p>
</blockquote>
<p>I started by stating that since $f$ is bounded, $\inf(f(x)) \le f(x) \le \sup(f(x))$. And then $|f(x... | Brethlosze | 386,077 | <p>From <a href="https://en.wikipedia.org/wiki/Line_(geometry)" rel="nofollow noreferrer">wikipedia</a> we can extract:</p>
<blockquote>
<p>Thus in differential geometry a line <em>may</em> be interpreted as a geodesic
(shortest path between points), while in some projective geometries a
line is a 2-dimensional ... |
3,336,135 | <p>I was looking at sequences <span class="math-container">$x_n=\{n\alpha\}$</span> with <span class="math-container">$n=1,2,\cdots$</span> and <span class="math-container">$\alpha\in [0, 1]$</span> an irrational number. Such sequences are known to be equidistributed, so they get arbitrarily close to any number <span c... | Vincent Granville | 574,948 | <p>So here is how it works. No proof here, but I am showing the pattern for all irrational numbers. </p>
<p><a href="https://i.stack.imgur.com/Bdpq5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Bdpq5.png" alt="enter image description here"></a></p>
<p>The above table is for the number <span clas... |
3,323,170 | <p>This question was originally posted <a href="https://crypto.stackexchange.com/q/72456/62225">here</a> on Crypto StackExchange. As suggested by an answer I am posting it here to help get a better perspective on the math side.</p>
<blockquote>
<p>Public-key cryptography was not invented until the 1970's. Apart from... | Wuestenfux | 417,848 | <p>My hypothesis is a bit different. From Euclid's ''Elements'' enough number theory (gcd, lcm) has been available to introduce (with some research) the theory of the RSA algorithm. The necessity to use cryptography can be seen from the Caesar cipher. Some handcrafting could have led to some larger primes and encryptin... |
645,579 | <p>Let $n$ be a positive integer. Find a general expression for $$\int x^n\cos(x)~dx$$ None of the standard integration techniques or the standard tricks I've seen for difficult integrals seem to apply to this one. I guess it is some type of reduction, but how to get a closed form?</p>
| Spock | 108,632 | <p>This integral is explicitly explained <a href="http://en.wikipedia.org/wiki/Integration_by_reduction_formulae" rel="nofollow">here</a>.</p>
|
645,579 | <p>Let $n$ be a positive integer. Find a general expression for $$\int x^n\cos(x)~dx$$ None of the standard integration techniques or the standard tricks I've seen for difficult integrals seem to apply to this one. I guess it is some type of reduction, but how to get a closed form?</p>
| Claude Leibovici | 82,404 | <p>Probably as you did, integrating twice by parts, you arrived to a simple recurrence relation between I(n) and I(n-2). You can check your results at<br>
<a href="http://en.wikipedia.org/wiki/Integration_by_reduction_formulae" rel="nofollow">http://en.wikipedia.org/wiki/Integration_by_reduction_formulae</a><br>
as men... |
214,475 | <p>Function:
<a href="https://i.stack.imgur.com/sH7mh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sH7mh.png" alt="enter image description here"></a></p>
<p>I am to solve for <span class="math-container">$T_{12}(4.8), T_{24}(1.2)$</span>, using <strong>If</strong> and <strong>Which</strong> funct... | Cesareo | 62,129 | <p>Try</p>
<pre><code>Clear[T, n, x]
Tn = T[n] /. RSolve[{T[n + 1] == 1/x T[n - 1] - 2/7 T[n], T[0] == 1, T[1] == x}, T, n][[1]]
Tn /. {n -> 12, x -> 4.8}
(* -0.0153785 *)
</code></pre>
|
2,071,332 | <p>I am working on implementation of a machine learning method that in part of the algorithm I need to calculate the value of $\Gamma (\alpha) / \Gamma (\beta) $. $\alpha$ and $\beta$ are quite large numbers (i.e. bigger than 200) and it causes the python $gamma$ function to overflow. However, as the difference of $\al... | Mark | 310,244 | <p>From <a href="https://math.stackexchange.com/questions/98348/how-do-you-prove-gautschis-inequality-for-the-gamma-function">this answer</a> we have Gautschi's inequality:</p>
<p>$$x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s},\quad x>0,0<s<1$$</p>
<p>We can combine this with the functional equa... |
4,058,319 | <p>My background in mathematical logic, model theory, etc. is patchy, so I'm looking for a clearer way to think about this. (Edit: I'm not asking what an isomorphism is, I'm asking how to formalize the idea of "preserving all logical properties" in order to state the principle described below in its full powe... | Noah Schweber | 28,111 | <p>Here is an outline of a theorem which, I think, addresses the situation pretty well (incidentally, <a href="https://mathoverflow.net/questions/336191/cauchy-reals-and-dedekind-reals-satisfy-the-same-mathematical-theorems">this MO question</a> is related):</p>
<p>Most logic books will include a proof that first-order... |
886,935 | <p>Let $A_1,A_2,A_3,\dots$ be a sequence of sequences where each
$$A_i = a_{i,1},a_{i,2},a_{i,3},\dots$$</p>
<p>Each sequence $A_i$ converges and in particular as $t \rightarrow \infty$, $a_{i,t} \rightarrow L_i$ for every $i$. We also have that in the limit as $i \rightarrow \infty$ the limits of these sequences con... | Ian | 83,396 | <p>Taking the theorem about limits of finite products for granted, you are asking the following: given $a_{i,j} \to L_i$ as $j \to \infty$ and $L_i \to 1$ as $i \to \infty$, when do we have</p>
<p>$$\lim_{j \to \infty} \lim_{k \to \infty} \prod_{i=1}^k a_{i,j} = \lim_{k \to \infty} \lim_{j \to \infty} \prod_{i=1}^k a_... |
3,148,094 | <p>In class, my professor computed the density of a gamma random variable by taking the derivative of its CDF, but he skipped many steps. I am trying to go through the derivation carefully but cannot reproduce his final result.</p>
<p>Let <span class="math-container">$k$</span> be the shape and <span class="math-conta... | Noah Caplinger | 581,608 | <p><span class="math-container">$A$</span> is a proper subset of <span class="math-container">$B$</span> if <span class="math-container">$B \subseteq A$</span> and there exists an <span class="math-container">$x \in A$</span> such that <span class="math-container">$x \notin B$</span>.</p>
<p>To show <span class="math-... |
1,557,039 | <h2>Background</h2>
<p>I am a software engineer and I have been picking up combinatorics as I go along. I am going through a combinatorics book for self study and this chapter is absolutely destroying me. Sadly, I confess it makes little sense to me. I don't care if I look stupid, I want to understand how to solve the... | lhf | 589 | <p>Actually, $3^{2n}\equiv 1\bmod 8$.</p>
<p>Indeed, by the binomial theorem, $3^{2n} = 9^n = (8+1)^n = 8a + 1$.</p>
|
109,961 | <p>Suppose $x^2\equiv x\pmod p$ where $p$ is a prime, then is it generally true that $x^2\equiv x\pmod {p^n}$ for any natural number $n$? And are they the only solutions?</p>
| André Nicolas | 6,312 | <p><strong>Way 1:</strong> Let us compute a bit. Let $p=2$. Note that $x^2\equiv x \pmod 2$ for any $x$. Is it true that always $x^2\equiv x\pmod 4$? No, let $x=2$.</p>
<p><strong>Way 2:</strong> We do more work, but will get a lot more information. Rewrite the congruence $x^2\equiv x \pmod p$ as $x^2-x\equiv 0 \pmod... |
102,624 | <p>I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one. </p>
<p>Since Maass cusp forms always are defined for a Fuchsian lattice, I... | Peter Humphries | 3,803 | <p>At the risk of blowing my own trumpet, I feel like I ought to mention a <a href="http://arxiv.org/abs/1502.06885" rel="nofollow">recent preprint of mine</a> that addresses this question.</p>
<p>Marc Palm answered your question 2, 3, and 4 reasonably well. (For question 4, the reference for where it is published is ... |
4,222,110 | <p>I am following course on topology that is kind of lack luster (not made for mathematicians). The course starts off with predicate logic and axiomatic set theory (ZFC). Now, I reached a point where the author defined the partition of unity and used the set of all continuous functions between 2 sets. But at the starts... | sirous | 346,566 | <p>An experimental approach:</p>
<p>Here I find one solution for this equation which a family of solutions can be based on:</p>
<p><span class="math-container">$9x^2=y^2-p=(y-\sqrt p)(y+\sqrt p)$</span></p>
<p>we can can construct following system of equations:</p>
<p><span class="math-container">$\begin{cases}y-\sqrt ... |
4,222,110 | <p>I am following course on topology that is kind of lack luster (not made for mathematicians). The course starts off with predicate logic and axiomatic set theory (ZFC). Now, I reached a point where the author defined the partition of unity and used the set of all continuous functions between 2 sets. But at the starts... | poetasis | 546,655 | <p>If we let <span class="math-container">$\quad A^2+B^2=C^2
\qquad A=3x\quad B=\sqrt{p}\quad C=y\qquad$</span>
we can use Euclid's formula for Pythagorean triples to find infinite solutions based on the value of <span class="math-container">$(x)$</span>. We start with the formula</p>
<p><span class="math-container">... |
96,657 | <p>I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous extension to $X$.</p>
<p>I think I know how to do this using Cauchy sequences, but there is this suggested alternativ... | Community | -1 | <p>Here is a perhaps an alternative way of looking at what happens before you take the infinite intersection. We know that $\textrm{diam} V_n(p) < \frac{2}{n}$. So by uniform continuity of $f$, for all $\epsilon > 0$ there exists $\delta > 0$ (and hence an $n\in \Bbb{N}$ such that $0 < \frac{1}{2n} < \de... |
41,302 | <p>This is a two part question, and for that I apologize.. but they're related!</p>
<p>Here's what I'm working with:</p>
<pre><code>d1 = Import["file.CSV", "List"]
size = Length[d1]
dis1 = RandomChoice[{d1}, {100, size}]
</code></pre>
<ul>
<li><p><strong>Q1</strong>: Length views <code>d1</code> as $300,000$ indiv... | DumpsterDoofus | 9,697 | <p>For Q1, your syntax mistake is an extra pair of brackets. The following code should work properly:</p>
<pre><code>d1 = Import["file.CSV", "List"]
size = Length[d1]
dis1 = RandomChoice[d1, {100, size}]
</code></pre>
|
320,228 | <p>Today, in my lesson, I was introduced to partial derivatives. One of the things that confuses me is the notation. I hope that I am wrong and hope the community can contribute to my learning. In single-variable calculus, we know that, given a function $y =f(x)$, the derivative of $y$ is denoted as $\frac {dy}{dx}$. I... | joriki | 6,622 | <p>First, rest assured that you're not the only one who's confused by the standard notation for partial derivatives. See <a href="https://math.stackexchange.com/a/101011">this answer</a> for a collection of answers I've written in response to such confusions.</p>
<p>The problem is that the standard notation doesn't in... |
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