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 |
|---|---|---|---|---|
1,000,547 | <p>So the system would look something like this.</p>
<pre><code>74" < 12.5x + 7.75y < 84"
60" < 12.5w + 7.75z < 74"
y + z = x + w
where x, y, w, z are natural numbers
</code></pre>
<p>Is this right? So i made a computer program and was able to solve the question by iterating through all the numbers. Ther... | Mark Bennet | 2,906 | <p>Well we could work in units of one quarter of an inch. The pieces we have are then $336$ units and $296$ units long respectively, a total of $632$ units.</p>
<p>The pieces we want to cut are $50$ units and $31$ units.</p>
<p>Now, we want equal numbers of the two sizes - so each pair comes to $81$ units.</p>
<p>$6... |
1,000,547 | <p>So the system would look something like this.</p>
<pre><code>74" < 12.5x + 7.75y < 84"
60" < 12.5w + 7.75z < 74"
y + z = x + w
where x, y, w, z are natural numbers
</code></pre>
<p>Is this right? So i made a computer program and was able to solve the question by iterating through all the numbers. Ther... | callculus42 | 144,421 | <p>The model, which is usually used is the following:</p>
<p>First you have to find which combinations of small pieces (12.5 inches and 7.75 inches) can be made out of the big pieces (84,74):</p>
<p><strong>84 Inches</strong></p>
<p>$$\begin{array}{|m{cm}|m{1cm}|}
\hline \text{combination} & 12.5 &7.75 &a... |
3,868,571 | <p>Let <span class="math-container">$n\ge 1$</span> and <span class="math-container">$A,B\in\mathrm M_n(\mathbb R)$</span>.</p>
<p>Let's assume that</p>
<p><span class="math-container">$$\forall Q\in\mathrm M_n(\mathbb R), \quad \det\begin{pmatrix} I_n & A \\ Q & B\end{pmatrix}=0$$</span></p>
<p>where <span cla... | Ben Grossmann | 81,360 | <p>Denote
<span class="math-container">$$M_Q = \begin{pmatrix} I_n & A \\ Q & B\end{pmatrix}, \quad \operatorname{col}(A,B) = \pmatrix{A\\B}.
$$</span></p>
<hr />
<p>The statement does hold with the additional assumption that <span class="math-container">$\ker(A) \subseteq \ker B$</span>, i.e. the row-space of ... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Stromael | 73,823 | <p>These answers all seem to be variations on one another, but I've found each one so far to be at least a little cryptic. Here's my version/adaptation.</p>
<p>Let <span class="math-container">$U \subseteq \mathbb{R}$</span> be open and let <span class="math-container">$x \in U$</span>. Either <span class="math-contain... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | RJM | 376,273 | <p>More of a question than answer. I am a chemist turned pharmacist, who wishes to have studied mathematics. I am trying to work through Rudin's Principles of Mathematical Analysis. I envy you all who are involved in math for a career. </p>
<p>Can someone give me feedback on my attempt at a proof in $\mathbb R? Com... |
2,056,209 | <p>We know that the union of countably many countable sets is countable.
What can se say about the union of infinitely many countable sets and its cardinality?
Thanks :)</p>
| Jacob Wakem | 117,290 | <p>The real numbers is the union of the singletons drawn from the real numbers. These are countable sets but the union is uncountable. In general the disjoint union of k (k greater than aleph null) countable sets (not the null set) has cardinality k.</p>
|
1,553,530 | <p>I have a small question about how to finish the proof in the title. The main idea seems to be make an assumption of ∀x∀y (Px→(Py→x=y)) and to derive a contradiction between Raa and ¬Raa from that, which then proves the conclusion:</p>
<p>So</p>
<p>1.∀x∀y (Px→(Py→x=y))</p>
<ol start="2">
<li>∀y (Pa→(Py→a=y))</li>
</o... | BrianO | 277,043 | <p>You don't know that $Pa$, hence the difficulty you're having.
I'll sketch the deduction:</p>
<ol>
<li>Assume $\forall x \forall y\,(Px\to (Py\to x=y))$.</li>
<li>Let $a$ be any individual. By hypothesis, we get</li>
<li>$\exists y\,(Ray \land Py)$. (This is your 6.) </li>
<li>$Rab \land Pb$ (existential elimination... |
1,553,530 | <p>I have a small question about how to finish the proof in the title. The main idea seems to be make an assumption of ∀x∀y (Px→(Py→x=y)) and to derive a contradiction between Raa and ¬Raa from that, which then proves the conclusion:</p>
<p>So</p>
<p>1.∀x∀y (Px→(Py→x=y))</p>
<ol start="2">
<li>∀y (Pa→(Py→a=y))</li>
</o... | Mario Carneiro | 50,776 | <p>My first thought when approaching this type of problem is to read out the statements and come up with a natural language proof. For convenience I will replace $Rxy$ with $x\prec y$ and read it as "$y$ is bigger than $x$" simply because we have a lot of convenient adjectives for that relation. $\forall x\exists y(x\p... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | known google | 392 | <p>EGA, with hyperlinks for easy navigation.</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Chern - Complex manifolds without potential theory</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Ricky | 7,845 | <p>All the SGA's. Note that SGA 1 and 2 already exists in TeX, and there is something for SGA 3 and 4.</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | hce | 2,781 | <p>Stong - Notes on cobordism theory</p>
|
79,658 | <p>My knowledge is very limited for complex geometry. I have the following question:</p>
<p>If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal O\left(E\right) \cong \mathcal O\left(F\right)$ between the sheaf of holomorphic sections, do we have an isomorphism $E \con... | DamienC | 7,031 | <p>Yes, this is true as soon as you assume that the isomorphism of sheaves $\mathcal O(E)\cong\mathcal O(F)$ is $\mathcal O_X$-linear. </p>
<p>To prove, pick a cover $(U_i)_{i\in I}$ of $X$ be opens which are trivializing both for $E$ and $F$. Now if you have an isomorphism of sheaves $\phi:\mathcal O(E)\to\mathcal O(... |
2,962,377 | <p>Rewrite <span class="math-container">$f(x,y) = 1-x^2y^2$</span> as a product <span class="math-container">$g(x) \cdot h(y)$</span> (both arbitrary functions)</p>
<p>To make more clear what I'm talking about I will give a example.</p>
<p>Rewrite <span class="math-container">$f(x,y) = 1+x-y-xy$</span> as <span class... | Batominovski | 72,152 | <p>For <span class="math-container">$f(x,y)=\dfrac{xy}{1-x^2y^2}$</span>, the answer is also no. Suppose on the contrary that there exist nonempty open intervals <span class="math-container">$U$</span> and <span class="math-container">$V$</span> such that <span class="math-container">$f|_{U\times V}$</span> can be fac... |
2,075,712 | <p>I want to show that if $A=[0,1)$ then its interior is $(0,1)$. I know that $int(A) \subset A$, and that $\forall x \in int(A)$ $\exists R>0 $ such that $B(x,R) \subset A$. Thus immediately we see that $0 \notin int(A)$ because $\not \exists R>0$ such that $B(0,R)\subset A$.</p>
<p>What I struggle to do is to ... | Community | -1 | <p>We know $$r = 4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$$ Thus, $$r_1-r = 4R\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2} -4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2} = 4R\sin ^2 \frac{A}{2}$$ Similarly it can be shown that: $$r_2-r = 4R\sin ^2 \frac{B}{2} \text{ and that } r_3-r = 4R\sin ^2\fr... |
1,940,249 | <p>According to Wikipedia, <a href="https://en.wikipedia.org/wiki/First-order_logic#Completeness_and_undecidability" rel="nofollow">first order logic is complete</a>. What is the proof of this?</p>
<p>(Also, in the same paragraph, it says that its undecidable. Couldn't you just enumerate all possible proofs and dispro... | Reese Johnston | 351,805 | <p>I won't cover why first order logic is complete - an adequate textbook on mathematical logic will provide the proof. I recommend <em>Mathematical Logic</em> by Kunen, but it may be a bit more extensive than your needs.</p>
<p>As for decidability: Every necessarily true statement is provable, and every necessarily f... |
3,061,277 | <p>This concept I have asked a few people, but none of them are able to help me understand, so hope that there's a hero can save me from this problem!!!</p>
<p>My question occurs during substitution process, for example, sometimes we let <span class="math-container">$x = π - u$</span>. Then after some manipulation of... | Doug M | 317,162 | <p>We have:</p>
<p><span class="math-container">$\int^{π}_{0} \frac{x\sin(x)}{(1+\cos^2(x))} dx = \int^{π}_{0} \frac{π\sin(u)}{(1+\cos^2(u))} du - \int^{π}_{0} \frac{u\sin(u)}{(1+\cos^2(u))} du$</span></p>
<p>Rather than say "let <span class="math-container">$u = x$</span>" I think it would be cleaner to say </p>
<p... |
556,664 | <p>For example, if $f$ is integrable on $[0,3]$, is it also integrable on $[1,2]$? I tried thinking of a counterexample but couldn't, since I've only learned what implies integrability but not what integrability implies.</p>
| Haha | 94,689 | <p>If $f$ is integrable on $[0,3]$ then there is a $F:[0,3]\to \Bbb R$ :$F'=f$ on $[0,3]$.The restriction of $F$ on $[1,2]$ let's name it $G$ means that $G'=f$ on $[1,2]$. So if $f$ is integrable on $[0,3]$ is integrable on every connected subset(<=>here is equal to every subinterval)
of $[0,3]$</p>
|
556,664 | <p>For example, if $f$ is integrable on $[0,3]$, is it also integrable on $[1,2]$? I tried thinking of a counterexample but couldn't, since I've only learned what implies integrability but not what integrability implies.</p>
| bof | 111,012 | <p>A function $f:[a,b]\to\mathbb R$ is Riemann-integrable if and only if $f$ is bounded and its set of points of discontinuity has Lebesgue measure zero. It clearly follow that, if $f$ is Riemann-integrable on an interval, then its restriction to a subinterval is also Riemann-integrable.</p>
|
1,407,169 | <p>Let $A$ and $B$ be two matrices in $M_n$.
Is the following ture:</p>
<p>$A$ and $B$ are similar $\iff$ $A$ and $B$ have the same jordan canonical form.</p>
<p>Could someone explain?</p>
| Patrick Stevens | 259,262 | <p>It's true. Let $A$ have the same JCF $J$ as $B$ does; say $A = PJP^{-1}$ and $B = Q J Q^{-1}$. Then $J = P^{-1} A P$, so $B = Q P^{-1} A P Q^{-1}$, and so $B$ is similar to $A$.</p>
<p>Suppose $A$ and $B$ are similar, but for contradiction they have different JCFs. Then $A$ reduces to JCF $J$, and $B$ to JCF $K$. B... |
2,338,199 | <p>Let $G$ and $H$ be two groups. Suppose that $M$ and $N$ be two normal subgroups of $G$ and $H$ respectively such that we have the following,</p>
<ul>
<li><p>$G$ and $H$ are isomorphic.</p></li>
<li><p>$M$ and $N$ are isomorphic.</p></li>
</ul>
<p>Then we know that $G/M$ and $H/N$ are isomorphic if for an isomorphi... | Ken Duna | 318,831 | <p>This is false. </p>
<p>Let $$G = \mathbb{Z}_2 \times \mathbb{Z}_2 = \textrm{ The Klein Four Group }$$ and $$M = \textrm{ trivial subgroup}.$$ Now let $$H = D_4 = \textrm{Dihedral Group of Order 8}$$ and $$N = Z(D_4) = \textrm{ The Center of } D_4.$$</p>
<p>You can show that $G/M \cong H/N$.</p>
|
3,764,342 | <p>Here since <span class="math-container">$\lim \frac{a_n}{a_{n+1}}=1.$</span> So no definite conclusion can be made about the nature of the sequence <span class="math-container">$\langle a_n\rangle$</span>.</p>
<p>So how can I can proceed to find the value of <span class="math-container">$a_1$</span> from the relatio... | Gerry Myerson | 785,985 | <p>When I was in high school a friend showed me a way to sum the infinite arithmetico-geometric series, made a big impression on me at the time: <span class="math-container">$$\matrix{1&+&x&+&x^2&+&x^3&+&x^4&+&\cdots\cr&&x&+&x^2&+&x^3&+&x^4&+&a... |
30,586 | <p>There are many ways to type a pipe. You could use \$|\$ (<span class="math-container">$|$</span>), \$\vert\$ (<span class="math-container">$\vert$</span>), \$\mid\$ (<span class="math-container">$\mid$</span>), or just a plain | (not surrounded by dollar signs). You could also use vmatrix to indicate matrix determin... | Martin Sleziak | 8,297 | <p>Since you asked about some specific meanings when the vertical line appears:</p>
<ul>
<li>For divisibility, you use <code>\mid</code> (and <code>\nmid</code>) for "does not divide". For example, <code>$a\mid b$</code> <span class="math-container">$a\mid b$</span>, <code>$4\mid8$</code> <span class="math-container">... |
3,831,310 | <p>I try to integrate</p>
<p><span class="math-container">$$\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$$</span></p>
<p>I did substituted <span class="math-container">$u = \frac{c}{m}$</span> and <span class="math-container">$w = g \sin \theta$</span> to get</p>
<p><span class="math-container">$$-\int \frac {dv}{... | DMcMor | 155,622 | <p>You've already rewritten the integral to make it easier to work with. The next step, as you've considered, is to try to get it into the form <span class="math-container">$\int \frac{dx}{1+x^2}$</span> so you can get an antiderivative in terms of arctangent. To do that:
<span class="math-container">\begin{align*}
-... |
3,696,265 | <p>This is from Vakil's FOAG: exercise 2.5 C, part b. I understand how objects in the extension sheaf from a sheaf on a base <span class="math-container">$\mathcal B$</span> of a topology are created, but I am having trouble understanding how to produce a morphism of sheaves given a morphism of sheaves on a base. </p>
... | Marko Riedel | 44,883 | <p>We seek to verify that</p>
<p><span class="math-container">$$\sum_{k=0}^{\min(p,q)} {p\choose k} (-1)^k {p+q-k\choose p} =1.$$</span></p>
<p>Re-write as</p>
<p><span class="math-container">$$\sum_{k=0}^{\min(p,q)} {p\choose k} (-1)^k {p+q-k\choose q-k}
\\ = [z^q] (1+z)^{p+q}
\sum_{k=0}^{\min(p,q)} {p\choose k} (-... |
2,368,771 | <blockquote>
<p>How to prove the function $$ f(z)=\exp\Big(\frac{z}{1-\cos z}\Big)$$ has an essential singularity at $z=0$ ?</p>
</blockquote>
<p>It's actually hard to express the Laurent series of $f(z)$ around $0$, because the power $\frac{z}{1-\cos z}$ itself is already in the series form (since $\cos z$ appears ... | smb3 | 467,465 | <p>We wish to show that the function's Laurent series has infinitely many negative terms. As you said, it may be difficult to express the entire Laurent series, but there is a formula for each of the Laurent coefficients:</p>
<p>$$a_n = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z^{n + 1}}dz$$</p>
<p>where $\gamma$ is ... |
743,911 | <p>Consider $\frac{1}{T}\sum_{t=1}^{T}\max\{ 0,a_t\}$. Can we say whether this is greater or equal then $\max\{ 0,\frac{1}{T}\sum_{t=1}^{T}a_t\}$?</p>
| Hayden | 27,496 | <p>Since $a_t\leq \max\{0,a_t\}$ for all $t$, it follows that $\frac{1}{T}\sum_{t=1}^T{a_t}\leq \frac{1}{T}\sum_{t=1}^T{\max\{0,a_t\}}$. Now, the RHS is always non-negative; if the LHS is non-negative, then it is equal to the max of $0$ and itself, and otherwise it is less than 0. In either case,
$$\max\left\{0,\fra... |
2,511,061 | <p>I will prove 0=1.
we know that, From Factorial definition
zero factorial is equal to one and
one factorial is equal to one.
so,0!=1!.
factorial get cancelled both sides,
we get 0=1.
Is this right..?</p>
| gt6989b | 16,192 | <p>What gives you the right to cancel out the factorial? By the same token I can claim that if $x^2 = y^2$ then $x=y$, which implies that since $1^2 = 1 = (-1)^2$, we must have $1 = -1$, just as absurd as your conclusion...</p>
|
2,511,061 | <p>I will prove 0=1.
we know that, From Factorial definition
zero factorial is equal to one and
one factorial is equal to one.
so,0!=1!.
factorial get cancelled both sides,
we get 0=1.
Is this right..?</p>
| Community | -1 | <p>Two things:</p>
<ol>
<li>"!" is not a variable rather it is an operator.</li>
<li>Factorial function (domain is whole number) is not injective.</li>
</ol>
|
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| mathreadler | 213,607 | <p><strong>Practical speed related aspects</strong> Matrix multiplication and division and equation system solving can become much faster to perform in some bases than others. Often one can cut the algorithmic complexity so much that it becomes a difference between something being practically useful or far too slow for... |
638,906 | <p>Find the coefficient of $x$ in the expansion of $$\left(1-2x^3+3x^5\right)\left(1+\frac{1}{x}\right)^5.$$ Answer is $154$, but how?</p>
| colormegone | 71,645 | <p>An alternate approach is to consider, without writing out the coefficients, that $ \ (1 + x^{-1})^5 \ $ will, from the binomial theorem, produce a polynomial of the form $ \ 1 + Ax^{-1} + Bx^{-2} + Cx^{-3} + Dx^{-4} + x^{-5} \ . $ When we multiply this by $ \ 1 - 2x^3 + 3x^5 \ , $ we see that multiplication by 1 ca... |
2,386,471 | <p>I have a solved question from Ross as stated below.</p>
<blockquote>
<p>Q : Suppose that each of three men at a party throws his hat into the
center of the room. The hats are first mixed up and then each man
randomly selects a hat. What is the probability that none of the three
men selects his own hat?</p>
... | Community | -1 | <p>For $n=1$, $\boldsymbol{c}=\boldsymbol0$ hence $\boldsymbol{a} = \boldsymbol0$.</p>
<p>For $n=2$, have you tried $$\boldsymbol C = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 ... |
2,386,471 | <p>I have a solved question from Ross as stated below.</p>
<blockquote>
<p>Q : Suppose that each of three men at a party throws his hat into the
center of the room. The hats are first mixed up and then each man
randomly selects a hat. What is the probability that none of the three
men selects his own hat?</p>
... | levap | 32,262 | <p>If $n = 1$, the only singular matrix is $C = 0$ but then any matrix $A$ will satisfy $AC = 0$ and not only $A = 0$.</p>
<p>For $n > 1$, denote by $e_1,\dots,e_n$ the standard basis of $\mathbb{F}^n$. Choose $1 \leq i \leq n$ and consider a matrix $C$ whose columns are the same and equal to $e_i$. This is a singu... |
365,166 | <p>I have trouble coming up with combinatorial proofs. How would you justify this equality?</p>
<p><span class="math-container">$$
n\binom {n-1}{k-1} = k \binom nk
$$</span>
where <span class="math-container">$n$</span> is a positive integer and <span class="math-container">$k$</span> is an integer.</p>
| Community | -1 | <p>Say we want to choose a baseball team of $k$ players from $n$ players and we want to choose a captain for the baseball team. We can count the above in two different ways.</p>
<p>$1$. Choose the captain first. This can be done in $n$ ways. Now choose the rest of the team, i.e., we need to choose $k-1$ people from th... |
86,277 | <p>I have a question: does the Heine-Borel theorem hold for the space $\mathbb{R}^\omega$ (where $\mathbb{R}^\omega$ is the space of countable sequences of real numbers with the product topology). That is, prove that a subspace of $\mathbb{R}^\omega$ is compact if and only if it is the product of closed and bounded su... | Nate Eldredge | 822 | <p>One thing you can say is that a subset of $\mathbb{R}^\omega$ is compact iff it is closed and <em>contained in</em> a product of bounded sets. I'll leave the proof as an exercise.</p>
<p>More generally, let $X_i$ be any family of Hausdorff spaces (and assume the axiom of choice). Then a subset $A$ of $X = \prod_i... |
287,405 | <p>Hi have this sequence:</p>
<p>$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}$$</p>
<p>I understand that this is a <em>Geometric series</em> so this is what I've made to get the sum.
$$\sum\limits_{n=1}^\infty (-1)^n\frac{3^{n}\cdot 3^{-2}}{4^n}$$
$$\sum\limits_{n=1}^\infty (-1)^n\cdot 3^{-2}{(\frac{3}{4})}^n... | amWhy | 9,003 | <p>The objective here is to transform your sum into a sum of the form: </p>
<p>$$\sum_{n=1}^\infty ar^{n-1}$$</p>
<p>$$\text{Transformation: }\quad\quad\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n} = \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\frac{(-3)^{n-1}}{4^{n-1}} = \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\left(... |
3,611,845 | <p>For the Stable Matching algorithm by Gale-Shapley, how do I prove that at most one man will get his worst choice? </p>
<p>My intuition is that I have to use contradiction. Assume that there are two men who will get their worst preferences: <span class="math-container">$M_1$</span> with <span class="math-container">... | Parcly Taxel | 357,390 | <p>Suppose <span class="math-container">$M_1$</span>'s last choice is <span class="math-container">$W_1$</span> and <span class="math-container">$M_2$</span>'s last choice is <span class="math-container">$W_2$</span>. Before they are forced to select <span class="math-container">$W_1$</span>/<span class="math-container... |
2,708,902 | <p>I am looking for an inductive proof of the following.</p>
<blockquote>
<p>Let $\alpha \in S_n$ be a cycle of length $p$. Then $\alpha^p =1$.</p>
</blockquote>
<p>For example $\alpha = (123)$ then $\alpha^3 = 1$ it is easy to verify.</p>
<p><em>My attempt:</em> Induction on $p$: The base case is true for $p=1$. ... | K B Dave | 534,616 | <p>I think the answer is <em>no</em> for the same reason that $x^p-x$ is not said to be the zero polynomial in $\mathbb{F}_p[x]$.</p>
<p>One "reason" that $x^p-x$ is not the zero polynomial in $\mathbb{F}_p[x]$ is that, even though it evaluates to zero in $\mathbb{F}_p$, it does not evaluate to zero in every $\mathbb{... |
77,246 | <p>What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
For example, is there any kind homology of orbifolds such that the first homology group is the abelianization of the ... | Angelo | 4,790 | <p>One can define singular homology directly for an orbifold, mimicking the standard constructions, but I don't know if this has been written up. The alternative is to take a space that is homotopy equivalent to your orbifold, and take the homology of that. This has been worked out by Behrang Noohi in <a href="https://... |
322,745 | <p>I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?</p>
<p>Let <span class="math-container">$d$</span> be a positive integer, <span class="math-container">$d\geq 2$</span>. By Dirichlet's theorem, there is an infinite set <span class="math-containe... | Greg Martin | 5,091 | <p>Chen's theorem says that there are infinitely many numbers <span class="math-container">$k$</span> such that <span class="math-container">$k-2$</span> is prime and <span class="math-container">$k$</span> is either prime or the product of two primes ("<span class="math-container">$k$</span> is a <span class="math-con... |
1,673,277 | <p>In what scenarios is Goldbach's conjecture, that all even numbers greater than 4 are the sum of two prime numbers; a natural conjecture to research?</p>
| Patrick Da Silva | 10,704 | <p>The point of such conjectures are often not the result themselves ; more often than not the knowledge of the veracity of such statements is completely uninteresting. The interest lies in the methods of proof to argue against such statements ; they often lead to developing new theories and inspire mathematicians to c... |
1,209,233 | <p>How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that one way of doing this is by brute force, but is there a more clever approach? Thanks in advance!</p>
| pjs36 | 120,540 | <p>We know subgroups of $S_4$ come in only a few different orders: $1,2,3,4,6,8,12,$ and $24$.</p>
<p>If we use the orbit stabilizer theorem, we have that $|H| = |\operatorname{Orb}_H(x)| \cdot |\operatorname{Stab}_H(x)|$; this limits us to only four possible subgroup orders, as $|\operatorname{Orb}_H(x)|$ can only be... |
701,681 | <p>It seems intuitive that, if a function differentiable on [a,b] is such that f'(a) < 0 < f'(b) then there exists some c in the open interval (a,b) such that f'(c)=0, but I can't prove it rigourously...</p>
<p>I expect to need to use Fermat's Theorem on stationary points but I would then need to prove that c is... | DonAntonio | 31,254 | <p>Hints:</p>
<p>$$f'(a)<0<f'(b)\implies \;\exists\,c,d\in (a,b)\;\;s.t.\;\;f\;\;\text{is descending in}\;\;[a,c)$$</p>
<p>$$\text{and ascending in}\;\;(d,b]$$</p>
<p>By continuity, there's some point in between (between <em>what</em>?) in which $\;f\;$ must have an extrema (in fact, a local minimum)</p>
|
269,696 | <p>I am given that $\sum\limits_{n=1}^\infty a_n$ is convergent. </p>
<p>I need to determine whether $\sum\limits_{n=1}^\infty (a_n)^\frac{1}{3}\;$ and $\;\sum\limits_{n=1}^\infty (a_n)^2\;$ are also convergent.</p>
<p>Imagine that $a_n = \dfrac{1}{n^4}.\;$ I believe that this is convergent because it's converging to... | amWhy | 9,003 | <p><strong>Task</strong>: You need to establish whether the statements are true <em>for all</em> $(a_n)$. </p>
<ul>
<li>To prove such a statement is <em>true</em>, you need to show it is <em>always holds</em> for any convergent $a_n$ (not just that it holds for <em>some</em> $a_n$). </li>
<li>But to conclude that a gi... |
1,080,746 | <p>I want to calculate the limit of this sum :</p>
<p>$$\lim\limits_{x \to 1} {\left(x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb\right)}$$</p>
<p>My efforts to solve the problem are described in the <a href="https://math.stackexchange.com/a/1308281/202081">self-answer below</a>.</p>
| Abdou Abdou | 202,081 | <p>We can define a rotation point when the curve switches its direction from increasing ordinates to decreasing values as x advances.</p>
<p><img src="https://i.stack.imgur.com/YlfWR.gif" alt="curve"></p>
<p>So when we try to locate those points in our polynomial, the derivative of this function must be 0 in that abs... |
2,649,557 | <p>I've been studying topology recently, and I've gotten to the part of the book that deals with quotient spaces. For the most part, it's fairly clear, but one thing that has been confusing me a bit is how the unit circle is represented.</p>
<p>Sometimes $\mathbf S^1$ is denoted as $\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)... | Berci | 41,488 | <p>These are different only on the level of <em>algebra</em>. <br>
From <em>geometric</em> point of view, these are exactly the same (with $w=a+bi$ correspondence), using the natural homeorphism $\Bbb R^2\cong\Bbb C,\ \ (x,y)\mapsto x+yi$.</p>
|
2,649,557 | <p>I've been studying topology recently, and I've gotten to the part of the book that deals with quotient spaces. For the most part, it's fairly clear, but one thing that has been confusing me a bit is how the unit circle is represented.</p>
<p>Sometimes $\mathbf S^1$ is denoted as $\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)... | William Elliot | 426,203 | <p>$S^1= \{ (x,y) \in R^2 : x^2 + y^2 = 1 \}.$<br>
All those other things are just homeomorphs, isomorphs even.<br>
Whether it is a subset of the real plain or the complex plain depends upon the context in use.</p>
|
294,895 | <p>Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces.</p>
<p>When is $\operatorname{Ran}(T)$ a closed subspace?</p>
<p>What theorems are there? </p>
<p>Thanks :) </p>
| Community | -1 | <p>A case in which it <em>is</em> true that $R(T)$ is closed is if $T$ is bounded from below, i.e., there exists $c > 0$ such that $\|Tx\| \geq c\|x\|$ for all $x \in X$. To prove this, notice first that since
$$Tx = 0 \implies 0 = \|Tx\| \geq c\|x\| \geq 0 \implies x= 0$$
$T$ must be injective. Now let $x_n \in R(T... |
4,383,098 | <p>This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are set, behave with some other operations defined on the set.</p>
<p>My question is how exactly do we find objects do f... | DanielWainfleet | 254,665 | <p>Convert to Boolean algebra: Let <span class="math-container">$I$</span> be any set such that <span class="math-container">$A\cup B\cup C\subseteq I.$</span> For any <span class="math-container">$S, T\subseteq I$</span> let <span class="math-container">$S+T=S\cup T$</span> and let <span class="math-container">$ST=S\t... |
1,958,616 | <p>Question: Let $z=x+iy $. Using complex notation, find an equation of a circle with radius $5$ and center $(3,-6) $.</p>
<p>The answer seems simple to derive, but I'm curious as to whether or not there is more that must be done to establish the equation than what I have accomplished.</p>
<p>The circle with the abov... | fleablood | 280,126 | <p>Definition of circle with radius $r$ centered at $(w,u)$: All points, $z$ of the complex plane where the distance from $z$ to $w + iu$ equals $r$. Or in other words: all $z$ where $|z - (w + ui)| = r$.</p>
<p>So the solution is $|z - (3 - 6i)| = 5$. That's all there is to it.</p>
<p>Now if $z = x+iy$ then $|z -... |
384,471 | <p>I'm attempting to evaluate the limit</p>
<p>$\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}$</p>
<p>I got it reduced to the following</p>
<p>$\lim_{x\rightarrow\infty}\frac{\sqrt{\frac{1}{\left(x-2\right)^{2}}-\frac{3}{\left(x-2\right)^{4}}}+1}{1-\frac{3}{\left(x-2\right)^{2}}-1}$</p>
<p>But putting in... | Easy | 60,079 | <p>\begin{align*}
&\lim_{x\rightarrow\infty}\frac{1}{\sqrt{x^{2}-4x+1}-x+2}\\
=&\lim_{x\rightarrow\infty}\frac{1}{\sqrt{(x-2)^2-3}-(x-2)}\\
=&\lim_{x-2\rightarrow\infty}\frac{1}{\sqrt{(x-2)^2-3}-(x-2)}\\
=&\lim_{z\rightarrow\infty}\frac{1}{\sqrt{z^2-3}-z}\\
=&\lim_{z\rightarrow\infty}\frac{\frac{1}{... |
1,915,257 | <p>$$\sum_{n=2}^\infty\frac{\cos\ln\ln n}{\ln n}$$
My idea is
$$-\frac1{\ln n}\le\frac{\cos\ln\ln n}{\ln n}\le\frac1{\ln n}$$
But I don't know if $\sum\frac1{\ln n}$ converges.</p>
| Mark Viola | 218,419 | <p>The first term in the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula#The_formula" rel="nofollow">Euler-Macluarin Summation Formula</a> is </p>
<p>$$\int_1^N \frac{\cos(\log\log x)}{\log(x)}\,dx=\int_{-\infty}^{\log\log N} e^{e^x}\cos(x)\,dx$$</p>
<p>which diverges as $N\to \infty$. Therefo... |
176,054 | <p>I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
Since I don't know the answer, I thought mathoverflow would be the right place to ask.
I mean, there are many areas o... | Manfred Weis | 31,310 | <p>Non-negative Matrix Factorization (cf e.g. <a href="http://en.wikipedia.org/wiki/Non-negative_matrix_factorization" rel="nofollow">http://en.wikipedia.org/wiki/Non-negative_matrix_factorization</a>) plays a role in knowledge representation and extraction.</p>
|
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | NereusF | 114,646 | <p>Refer <a href="http://www.dspguide.com/CH32.PDF" rel="noreferrer">http://www.dspguide.com/CH32.PDF</a> for an excellent explanation of Laplace Transforms in the Electrical Domain.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Amir Asghari | 29,316 | <p><a href="http://books.google.com/books/about/Mathematics_A_Very_Short_Introduction.html?id=DBxSM7TIq48C">Mathematics: A Very Short Introduction</a> by Timothy Gowers. It is very short and indeed very insightful. It is not a textbook, but includes some school-mathematics topics. From the cover: </p>
<blockquote>
<... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Gerry Myerson | 3,684 | <p>Walter Prenowitz and Meyer Jordan, <em>Basic Concepts of Geometry</em>. </p>
|
1,974,939 | <p>I don't understand red line parts. I dont know why the assumtion that the list in (1) contains every real number leads to the conclusion that the intersection n to infinite number In equal to empty set.
Please explain why... I will really appreciate it. </p>
<p><a href="https://i.stack.imgur.com/UtJHG.png" rel="no... | angryavian | 43,949 | <p>Right before the part that you underlined, they show that if $x$ is in that list (1), then it is not in the infinite intersection $\bigcap_{n=1}^\infty I_n$. By construction, the list (1) contains all real numbers, so all real numbers are not in $\bigcap_{n=1}^\infty I_n$, or in other words, this infinite intersecti... |
1,858,218 | <p>I often come across students who are confused by the idea that the complex unit, $i$, is defined as $i^2 = -1$. Since we are using the complex numbers in an engineering course, we use the complex numbers to encode information about a sinusoid of a given frequency, where the argument is the phase of the sinusoid wit... | Community | -1 | <p>An alternative to the use of the embarrassing $i=\sqrt{-1}$ is to work with coordinate couples $(a,b)$ and introduce the product rule</p>
<p>$$(a,b)\cdot(c,d):=(ac-bd,ad+bc).$$</p>
<p>Then the constant $i$ is the innocuous $(0,1)$.</p>
<hr>
<p>An important issue is to relate the complex product to rotations. It ... |
11,368 | <p>I am interested in improving this plot:</p>
<p><img src="https://i.stack.imgur.com/qvPAJ.png" alt="contour plot"></p>
<p>which I produced with the following command:</p>
<pre><code>dat // ListContourPlot[#, ContourShading -> False,
ContourStyle -> ColorData[10] /@ Range[10], Contours -> Range[21]/8,
... | rm -rf | 5 | <p>Here's my solution. Step-by-step explanation follows.</p>
<p><img src="https://i.stack.imgur.com/D900n.png" alt="final solution"></p>
<ul>
<li><p>First, import the data</p>
<pre><code>dat = ToExpression@Import["http://pastebin.com/raw.php?i=XWyb7jFJ"];
</code></pre></li>
<li><p>Instead of using <code>EdgeDetect</... |
4,412,175 | <p>I wonder and tried to google it, but I am not sure what to google it, how to solve non linear equations where equations are equal between each other. I am able to write a specific algorithm for two equations but not dynamically for N equations. I will show the example of three (how my equations approximately looks l... | Jean Marie | 305,862 | <p>A much more direct way is to consider that this kind of system is composed of an eigensystem condition + (the last equation which is plainly a normalizing condition.</p>
<p>Let us take your example written under the form of a <em>slightly simplified</em> system</p>
<p><span class="math-container">$$\begin{cases}2C_1... |
4,038,802 | <p>Say I have <span class="math-container">$2^n +1 < n! -n$</span> for all <span class="math-container">$n \ge 4$</span>, and <span class="math-container">$n$</span> is an integer.</p>
<p>My inductive steps says, consider a <span class="math-container">$k$</span> that is a arbitrary integer, assuming <span class="ma... | Community | -1 | <p>You can get all the transpositions from those three. There are <span class="math-container">${4\choose2}=6$</span> transpositions total, so there are sort of <span class="math-container">$3$</span> things to check.</p>
<p>For instance, <span class="math-container">$(12)(23)(12)=(13)$</span>.</p>
<p>(One more hint: ... |
2,960,614 | <p>I am going through Tenenbaum and Pollard's book on differential equations and they define the differential <span class="math-container">$dy$</span> of a function <span class="math-container">$y = f(x)$</span> to be the function <span class="math-container">$$
(dy)(x,\Delta x) = f'(x) \cdot (d\hat{x})(x, \Delta x)
$$... | Hir | 506,981 | <p>That holds.
You get to be so careful about where outer measure is defined on.
We start with a measurable space (X,F) and let Γ be an outer measure on (X,F) and let μ be an measure on it.
Let's go through what we've learned.</p>
<p>Γ: 2^X→[0,+∞] holds
Γ(∪i E_i) ≦ ΣiΓ(E_i) (subadditivity) as for E_i∈F i=1,2,..., rega... |
2,395,611 | <p>This is probably an easy question, hanging on some minor detail, but i cannot find a proof for it. I try working through T.A. Springer's book "Linear algebraic groups" and i got stuck at remark 7.1.4. (I know there is already a question asked about the same remark, but concerning a different statement made in it). S... | D_S | 28,556 | <p>Check that $N_{G/S}(T/S) = N_G(T)/S$ and $Z_{G/S}(T/S) = Z_G(T)/S$ and use the third isomorphism theorem.</p>
<p>I'll write out the details of why $Z_{G/S}(T/S) = Z_G(T)/S$. The inclusion '$\supseteq$' is clear. Conversely, let $gS \in G/S$, and suppose $gS$ commutes with every element of $T/S$. This means that ... |
1,249,638 | <p>Update: This is false. See the answers for a counterexample.</p>
<blockquote>
<p>Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p
\right)^{1/p}$$ is non-decreasing.</p>
</blockquote>
<p>Comments: I'm posting this because there is (surprisingly) no go... | Ragnar | 232,420 | <p>Jensen:</p>
<p>take q>p, examine:</p>
<p>$L(q,p)$ = $\frac {(\int |f|^q)^{\frac 1q}}{(\int |f|^p)^{\frac 1p}}$</p>
<p>By Jensen's inequality $(\int |f|^p)^{\frac qp}$ ≤ $(\int |f|^q)$ Because $\frac qp$≥1 $\Rightarrow$ $g(x)$ = $x^{\frac qp}$ is convex</p>
<p>Thus, $L(q,p)^q$≥ $\frac {\int |f|^q}{\int |f|^q}$ = ... |
1,249,638 | <p>Update: This is false. See the answers for a counterexample.</p>
<blockquote>
<p>Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p
\right)^{1/p}$$ is non-decreasing.</p>
</blockquote>
<p>Comments: I'm posting this because there is (surprisingly) no go... | Jose27 | 4,829 | <p>This is not true: Consider the function $f(x)=\chi_{\mathbb{R}\setminus(-1,1)}(x)|x|^{-1/2}$. Then for $p\leq 2$ we have $\| f\|_{p}=\infty$ and for $p>2$ we get
$$
\| f\|_p = \left(\frac{4}{p-2}\right)^{1/p}.
$$</p>
|
1,176,016 | <p>How do you show that if you have sets $B_1, B_2, \cdots ,B_n$ and a set $C$, then
$$(B_1\cap B_2, \cap \cdots B_n)\cup C= (B_1\cup C)\cap(B_2\cup C) \cap \cdots
\cap (B_n\cup C)\,?$$</p>
<p>Thanks</p>
| kingW3 | 130,953 | <p>Logically derivatives represent the rate of change of the function,the rate of change doesn't take into consideration from what value the function starts to change at that rate.</p>
<p>A simple example is $2x+15=2x$.</p>
<p>Now the other way around if two functions take the same value at one point they don't have ... |
3,066,991 | <p>I'm working through some functional analysis problems and am having trouble with the following. Let <span class="math-container">$f(z)=\sum_{n=0}^\infty a_n z^n$</span> be a power series with radius of convergence <span class="math-container">$R>0$</span>. Let <span class="math-container">$A\in\mathcal{B}(\mathc... | Martin Argerami | 22,857 | <p>In part a), your inequality is too crude. And it wouldn't prove much, because what you need to show is that the series exists. You have
<span class="math-container">$$
\left\|\sum_{n=m}^ka_nA^n\right\|\leq\sum_{n=m}^k|a_n|\,\|A^n\|\leq\sum_{n=m}^k|a_n|\,\|A\|^n.
$$</span>
Since <span class="math-container">$\|A\|&l... |
3,066,991 | <p>I'm working through some functional analysis problems and am having trouble with the following. Let <span class="math-container">$f(z)=\sum_{n=0}^\infty a_n z^n$</span> be a power series with radius of convergence <span class="math-container">$R>0$</span>. Let <span class="math-container">$A\in\mathcal{B}(\mathc... | mechanodroid | 144,766 | <p>Define <span class="math-container">$T = f(A)$</span> with <span class="math-container">$f(A) = \sum_{n=0}^\infty a_nA^n$</span>. This series converges absolutely since
<span class="math-container">$$\sum_{n=0}^\infty |a_n|\|A^n\| \le \sum_{n=0}^\infty |a_n|\|A\|^n < +\infty$$</span>
because <span class="math-con... |
2,727,598 | <p>Given p is a prime number greater than 2, and</p>
<p>$ 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{p-1} = \frac{N}{p-1}$ </p>
<p>how do I show, $ p | N $ ???</p>
<p>The previous part of this question had me factor $ x^{p-1} -1$ mod $p$. Which I think is just plainly $(x-1) ... (x-(p-1))$ </p>
| Piquito | 219,998 | <p>HINT.-$x\to\dfrac 1x$ is a bijection of $\mathbb F^*_p$ so $$\sum_1^{p-1} \frac 1x\equiv\sum_1^{p-1} x\equiv 0\pmod p$$</p>
|
476,955 | <p>Let $X$ be a set with 4 elements. Is it possible to have a topology on $X$ with 14 open sets?</p>
| Ittay Weiss | 30,953 | <p>Hints:</p>
<p>1) If each singleton is open, then the topology is the discrete one. So assume not every singleton is open. </p>
<p>2) Since we are allowed to miss exactly two subsets from the list of open sets, and there are four singletons, without loss of generality, $\{1\}$ and $\{2\}$ are open.</p>
<p>3) Of th... |
606,380 | <blockquote>
<p>If <span class="math-container">$abc=1$</span> and <span class="math-container">$a,b,c$</span> are positive real numbers, prove that <span class="math-container">$${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1\,.$$</span></p>
</blockquote>
<p>The whole problem is in the title. If you wanna h... | math110 | 58,742 | <p>let $$a=x^3,b=y^3,c=z^3\Longrightarrow xyz=1$$
since
$$y^3+z^3\ge y^2z+yz^2$$
so
$$\dfrac{1}{1+b+c}=\dfrac{xyz}{xyz+y^3+z^3}\le\dfrac{xyz}{xyz+y^2z+yz^2}=\dfrac{x}{x+y+z}$$
so
$$\sum_{cyc}\dfrac{1}{1+b+c}\le\sum_{cyc}\dfrac{x}{x+y+z}=1$$</p>
|
2,177,619 | <p>Find a solution to the boundary value problem \begin{align}y''+ 4y &= 0 \\ y\left(\frac{\pi}{8}\right) &=0\\
y\left(\frac{\pi}{6}\right) &= 1\end{align}
if the general solution to the differential equation is $y(x) = C_1 \sin(2x) + C_2 \cos (2x)$.</p>
<p>I was able to compute the following equations:
\b... | Bob | 154,155 | <p><span class="math-container">\begin{align*}
\frac{\sqrt 2 C_1}{2} + \frac{\sqrt 2 C_1}{2} &= 0 \\
C_1 + C_2 &= 0 \\
C_2 &= -C_1 \\
\frac{\sqrt{3}C_1}{2} + \frac{C_2}{2} &= 1 \\
\sqrt{3}C_1 + C_2 &= 2 \\
\sqrt{3}C_1 - C_1 &= 2 \\
(\sqrt{3} - 1)C_1 &= 2 \\
C_1 &= \frac{2}{(\sqrt{3} - 1)... |
3,645,742 | <p>The number <span class="math-container">$2012 \cdot 2013 \cdot 2014 + 2013$</span> is the cube of </p>
<p>a) <span class="math-container">$2012$</span><br>
b) <span class="math-container">$2013$</span><br>
c) <span class="math-container">$2014$</span><br>
d) <span class="math-container">$2112$</span><br>
e) <span c... | qwr | 122,489 | <p>You can also guess the answer by estimation and the process of elimination.</p>
<p><span class="math-container">$2012^3$</span> is too small because <span class="math-container">$2012 \times 2013 \times 2014$</span> is already larger.</p>
<p><span class="math-container">$2014^3$</span> is too large because <span c... |
57,508 | <p>While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question: </p>
<p>Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomia... | Donu Arapura | 4,144 | <p>Here's a short argument over $\mathbb{C}$.</p>
<p>If $\mathbb{A}^2-\{0\}$ were affine, then a standard application of Morse theory [Milnor, chap 1, sect 7] would show that $H^i(\mathbb{A}^2-\{0\},\mathbb{Z})=0$ for $i>2$. But it's homotopic to $S^3$.</p>
|
57,508 | <p>While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question: </p>
<p>Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomia... | Brian | 9,035 | <p>We can easily see that the function field of $\mathbb{A}^2_k-(0,0)$ is still $k(x,y)$. So the ring of functions is of the form $f/g$ where $f$ and $g$ are polynomials. But any polynomial in 2 variables will vanish at a codim 1 sub-variety, i.e. cannot vanish at exactly 1 point. This is the Krull dimension theorem. B... |
2,939,058 | <p>The question to be solved is:</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \left( \ \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} \ - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \ \right)$$</span></p>
<p>The first thing that occured to me was to transform the limits into definite inte... | Jose Arnaldo Bebita Dris | 28,816 | <p><strong>MY ATTEMPT</strong></p>
<p>For even perfect numbers <span class="math-container">$2^{p-1}(2^p - 1)$</span>, I get
<span class="math-container">$$D(2^p - 1)D(2^{p-1}) = (2(2^p - 1) - (2^p))(1) = 2^{p+1} - 2 - 2^p = 2^p - 2$$</span>
<span class="math-container">$$2s(2^p - 1)s(2^{p-1}) = 2(2^p - (2^p - 1))(2^... |
1,106,320 | <p>I know the chance of hitting either red or black on the first roll is about 48%
1/37 chance on European 0
But shouldn't the chance of getting it right on either the first or second roll be high</p>
<p>like 66% or 75%</p>
<p>so to bet 25 1st roll and 50 second roll should have about 75% chance of a win?</p>
| user208259 | 208,259 | <p>If the chances of winning on one roll are $\frac{18}{37}$, then the probability of winning <em>at least once</em> in two rolls is
$$1 - (1 - 18/37)^2 \approx 74\%.$$</p>
|
1,968,267 | <p>When doing induction should you always try to put your final answer as the "<em>desired</em> " form? For example if: $$\sum^{n}_{k=1}(k+2)(k+4) = \frac{2n^{3} + 21n^{2} + 67n}{6}$$ we ought to give the final answer as $$\frac{2(k+1)^{3} + 21(k+1)^{2} + 67(k+1)}{6}?$$</p>
<p>I just expanded both the $\text{LHS}_{k+1... | ypercubeᵀᴹ | 7,367 | <p>You could also use that: </p>
<p>$$(k+2)(k+4) = (k+3)^2 - 1$$</p>
<p>and the known sum:</p>
<p>$$ \sum^{n}_{k=1}{k^2} = \frac{n(n+1)(2n+1)}{6} $$</p>
<p>$$ \sum^{n}_{k=1}{(k+2)(k+4)} = \sum^{n}_{k=1}{((k+3)^2 - 1)} = \sum^{n}_{k=1}{(k+3)^2} - \sum^{n}_{k=1}{1} $$
$$ \sum^{n+3}_{k=4}{k^2} - n = \sum^{n+3}_{k=1}{k... |
2,381,496 | <p>I am trying to write the explicit formula of all solutions of a linear system in the form :</p>
<p>$Ax=b$ where $A$ is an $m \times n$ matrix ($n$ different from $m$), $x$ is $n$-dimensional vector and $b$ is $m$-dimensional vector.</p>
| platty | 468,546 | <p>Rewrite the first equation as $c = \frac{ab}{4} - a - b$. Square it to get $$c^2 = a^2 + b^2 + \frac{a^2b^2}{16} - \frac{a^2b}{2} - \frac{ab^2}{2} + 2ab$$
Now using the other equation, we see that
$$\frac{a^2b^2}{16} - \frac{a^2b}{2} - \frac{ab^2}{2} + 2ab = 0$$
Since $a,b > 0$ divide by $ab$ and multiply by $16$... |
2,381,496 | <p>I am trying to write the explicit formula of all solutions of a linear system in the form :</p>
<p>$Ax=b$ where $A$ is an $m \times n$ matrix ($n$ different from $m$), $x$ is $n$-dimensional vector and $b$ is $m$-dimensional vector.</p>
| fleablood | 280,126 | <p>Hint: formula for primitive triples is </p>
<p>$a = mn; b = \frac {m^2 -n^2}2; c = \frac {m^2 + n^2}2$ for $\gcd(m,n) = 1$. And as $m^2 - n^2 $ is even $m$ and $n$ must both be odd.</p>
<p>So we want $\frac {mn(m^2 -n^2)}4 = 2(mn + \frac {m^2 - n^2}2 + \frac {m^2 - n^2} 2) = 2(mn + m^2)$</p>
<p>So $mn(m-n)(m+n) =... |
202,530 | <p>I'd like to check if all elements of a list are numbers. I've tried </p>
<pre><code>t = {5/4, 12}
MatrixQ[t, NumberQ]
MemberQ[t, NumberQ]
And @@ Table[NumberQ[t[[i]]], {i, 1, Length[t]}]
</code></pre>
<p>but only the last one yields the desired result. Is there a better way to check?</p>
| mgamer | 19,726 | <p>Given a list:</p>
<pre><code>list = {Pi, 0.4, 1, 2/2, 1./3}
</code></pre>
<p>you can do:</p>
<pre><code>And @@ (Head[#] === Real & /@ list)
(* False *)
</code></pre>
|
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| Empy2 | 81,790 | <p>$$5^\sqrt{3}>5^{5/3}=\sqrt[3]{3125}>\sqrt[3]{2187}=3^{7/3}>3^{\sqrt{5}}$$<br>
Another option is to take roots, and compare $3^{1/\sqrt{3}}$ with $5^{1/\sqrt{5}}$. This function increases for $1<x<7.39$, then decreases.</p>
|
972,385 | <p>Could someone please explain this question to me? I know that such integers do NOT exist, but I could not prove it. "Either solve it or give a brief explanation as to why it is impossible." Thank you!</p>
<p>Find integers $s$ and $t$ such that $15s + 11t = 1$.</p>
| Lucian | 93,448 | <p>$$15s+11t=1\iff11(s+t)+4s=1\iff4~\Big[3(s+t)+s\Big]-(s+t)=1=4-3.$$ Now let $s+t=3$ and $4s+3t=1$, which is a system of two equations in two unknowns, yielding $s=-8$ and $t=11$. Then it is trivial to show that all integers of the form $s+11n$ and $t-15n$ are also solutions to the initial equation.</p>
|
1,839,458 | <blockquote>
<p>For any $n\ge2, n \in \mathbb N$ prove that
$$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$</p>
</blockquote>
<h3>My work so far:</h3>
<p>1) $$\sqrt{n+1}-\sqrt{n}>\frac1{2\sqrt{n+0.5}}$$</p>
<p>2) $$\sqrt{n+1}-\sqrt... | Jack D'Aurizio | 44,121 | <p>We have to prove that:
$$ \sum_{k=1}^{2n}\sqrt{2k-1}-\sum_{k=1}^{2n}\sqrt{2k-2} > \sqrt{n} \tag{1}$$
hence it looks like a good idea to apply creative telescoping and approximate:
$$\sqrt{2k-1}-\sqrt{2k-2}\geq\frac{\sqrt{k-1/4}-\sqrt{k-5/4}}{\sqrt{2}}-\frac{1}{128\sqrt{2}}\left(\frac{1}{(k-5/4)^{3/2}}-\frac{1}{(k... |
3,161,584 | <p>Consider the following matrix</p>
<p><span class="math-container">$A=\begin{bmatrix}0&0&0&0\\a&0&0&0\\0&b&0&0\\0&0&c&0\end{bmatrix}$</span></p>
<p>Which <span class="math-container">$a,b,c$</span> are real numbers</p>
<p>What conditions are required for <span class=... | coding1101 | 633,698 | <p>Thanks @Blue.</p>
<p>I solved this problem in a very simple way, thanks to you.</p>
<p>1) Solved this by contradiction.</p>
<p>If <span class="math-container">$|M| \ge 4$</span>, the lengths of edges of quadrangles are all different.</p>
<p>Let the four lengths <span class="math-container">$a, b, c, d$</span>.</... |
1,809,500 | <p>Prove that $k!>(\frac{k}{e})^{k}$. </p>
<p>It is known that $e^{k}>(1+k)$. So if we multiply $k!$ on both sides, we get $k!e^{k}>(k+1)!$. Also $k^k>k!$. Now how to proceed ?</p>
| Chris Sanders | 309,566 | <p>I take it that $k$ is a positive integer. Then the question is equivalent to</p>
<p>$e^k>\frac{k^k}{k!}$</p>
<p>But $e^k=\displaystyle \sum_{n=0}^\infty \frac{k^n}{n!}$ and one of the summands (all of them are positive) is itself $\frac{k^k}{k!}$</p>
|
1,876,590 | <p>In "<a href="https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf" rel="nofollow">Angle Trisection, the Heptagon, and the Triskaidecagon</a>", published in the <em>American Mathematical Monthly</em> in March 1988, Andrew Gleason discusses what regular poly... | Oscar Lanzi | 248,217 | <p>If the object is to construct a regular hendecagon, it can be done more simply without going through an angle quinsection. Benjamin and Snyder proved the existence of a construction using a marked ruler and compasses in 2014 (BENJAMIN, ELLIOT; SNYDER, C. Mathematical Proceedings of the Cambridge Philosophical Socie... |
637,699 | <p>We have a triangle $ABC$. Whats the value of angle $C$?</p>
<p>$$\sin^2(A)+\sin^2(B)-\sin^2(C)=1$$</p>
<p>I made a small java program and it gave me an answer. I want to know how to make it through other ways.</p>
| JB King | 8,950 | <p>An isosceles triangle with angles of 90,45 and 45 comes to mind as a rather simple solution. Let A or B be one of the 90 degree angles since $sin^2(90)=1$ and the others will cancel out. Whether there is another solution one may use substitution since A+B+C=180.</p>
|
637,699 | <p>We have a triangle $ABC$. Whats the value of angle $C$?</p>
<p>$$\sin^2(A)+\sin^2(B)-\sin^2(C)=1$$</p>
<p>I made a small java program and it gave me an answer. I want to know how to make it through other ways.</p>
| Blue | 409 | <p>As @lab shows in a comment,
$$\sin^2 A + \sin^2 B - \sin^2 C = 1 \quad\implies\quad 2\sin A \sin B \cos C = 1$$
so we can write
$$1 = 2 \sin A \sin(A+C) \cos C = 2 \sin A \cos C( \sin A \cos C+ \cos A \sin C)$$
Thus,
$$\begin{align}
1 - 2 \sin^2 A \cos^2 C &= 2 \sin A \cos A \sin C \cos C \\
\implies \quad \left... |
50,406 | <p>This is a question about support of modules under extension of scalars.</p>
<p>Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module.
Recall that the <em>support</em> of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localizat... | jdc | 5,792 | <p>Okay, a further question. Despite its simplicity, it is not homework.</p>
<p>Atiyah and Macdonald (Ch. 2, Ex. 19.iv))
claim that if $M$ is a sum of submodules $M_i$,
then $\text{Supp}(M) = \bigcup \text{Supp}(M_i)$,
not specifying what size of set the indices $i$ are to run over.
I assumed initially the index se... |
4,218,944 | <p>A very simple question. Saw this formula many places earlier, but how do we prove it?
<span class="math-container">$$ax^2+bx+c=a(x-r_1)(x-r_2)$$</span> Where <span class="math-container">$r_1$</span> and <span class="math-container">$r_2$</span> are the roots of the quadratic.</p>
<p>P.S.- I have seen a "proof&... | Deepak | 151,732 | <p>What do you define as "roots" of the quadratic? If they are precisely the two (not necessarily distinct) values that make the quadratic zero, then it's as trivial as setting the factorised expression to zero and deducing that one or the other linear factor has to be zero. The two roots of the LHS correspon... |
3,623,856 | <p>Use the definition of a Cauchy sequence to prove that the sequence defined by <span class="math-container">$x_n = \left (\frac{3}{2}\right )^n$</span> is a Cauchy sequence in <span class="math-container">$\mathbb{R}$</span>. </p>
| Saaqib Mahmood | 59,734 | <p>Your sequence is <em>not</em> a Cauchy sequence. </p>
<p>In order to show this, let <span class="math-container">$m$</span> and <span class="math-container">$n$</span> be any natural numbers such that <span class="math-container">$m <n$</span>. Then we note that
<span class="math-container">$$
\begin{align}
\lef... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Yuval Filmus | 1,277 | <p>The most interesting aspect for me is the "elementary puzzle questions", questions which are natural, require some thought, and are elementary in nature. In that respect, the advantage of Math.SE over MathOverflow is that the questions are actually elementary, whereas in MO they usually involved subjects I know noth... |
18,421 | <p>I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concrete... | Andrew Chin | 12,720 | <p>Use a <em>language</em> approach.</p>
<p>This is not meant to be a snide remark; I came across a <a href="https://www.youtube.com/watch?v=V6yixyiJcos" rel="nofollow noreferrer">particular TED talk</a> a while ago that illustrated how math is just another language and we must first learn the proper syntax.</p>
<p>O... |
804,310 | <p>The probability distribution of a discrete random variable x is
$$f (x)= \begin{pmatrix}3 \\ x \end{pmatrix} (1/4)^x (3/4)^{3-x} $$ Find the mean value of x.
Construct a cumulative distribution function for f (x).
i find out$$ P(X=o) = 0.421875$$ $$ P(X=1) = 0.421875$$ $$ P(X=2) = 0.140625$$ $$ P(X=3) = 0.015... | Bowen | 101,918 | <p>This discrete random variable follows Binomial distribution, i.e.
$$f(x)=\binom{n}{x}p^x(1-p)^{n-x}$$
where $n$ is the number of trials and $p$ is the proportion. What this distribution mean is that when there are $n$ trials and every trial follows $p$ proportion(success rate), the probability that $x$ successes occ... |
1,087,910 | <p>Erdős proved that if $f(n)$ is a monotone increasing function from the natural numbers to the positive reals, and $f(n)$ is completely multiplicative, then there exists some constant $C$ such that $f(n)=n^C$ for all $n$.</p>
<p>I have not been able to find a nice proof of this online and was hoping someone could pr... | Huy | 3,787 | <p>Using partial fractions, you'll find</p>
<p>$$\frac{1}{x^2(x-1)^3} = -\frac{1}{x^2} - \frac{3}{x} + \frac{3}{x-1} - \frac{2}{(x-1)^2} + \frac{1}{(x-1)^3}.$$</p>
<p><strong>EDIT:</strong> To integrate the fractions in your original post you claim to have trouble with, remember that</p>
<p>$$\left(\frac{1}{x} \righ... |
3,369,129 | <p>I have the following formulas: </p>
<p><span class="math-container">$$S = 4 \times \tan\left(\frac{180^\circ-a}{2}\right) - \frac{π}{90^\circ} \times (180^\circ-a)$$</span></p>
<p>where:</p>
<p><span class="math-container">$$S \gt 0$$</span></p>
<p><span class="math-container">$$0^\circ \lt a \lt 180^\circ$$</... | Lutz Lehmann | 115,115 | <p>You also have, in the relation of degree and radian units, <span class="math-container">$\pi=180^∘$</span>, <span class="math-container">$90^∘=\frac\pi2$</span> etc. so that the equation reduces to
<span class="math-container">$$
\frac{S}4=\tan(x)-x
$$</span>
where <span class="math-container">$x=\frac{180^∘−α}2$<... |
358,328 | <p>How do I go about doing this? I'm clueless..
Thank you.</p>
<p>My attempt:</p>
<p>Using the product rule and making:</p>
<p>$\eqalign{
& u = u \cr
& v = {v^{ - 1}} \cr} $</p>
<p>so:</p>
<p>${{du} \over {dx}} = 1$
and ${{dv} \over {dx}} = - {v^{ - 2}}$
so:</p>
<p>$\eqalign{
& {{dy} \over {d... | Community | -1 | <p>We have
$$\left(\frac{1}{v}\right)'=-\frac{v'}{v^2}$$
and $$(ut)'=u't+ut'$$
then we take $t=\frac{1}{v}$ and we find
$$\left(\frac{u}{v}\right)'=\frac{u'}{v}-\frac{uv'}{v^2}=\frac{u'v-uv'}{v^2}$$</p>
|
358,328 | <p>How do I go about doing this? I'm clueless..
Thank you.</p>
<p>My attempt:</p>
<p>Using the product rule and making:</p>
<p>$\eqalign{
& u = u \cr
& v = {v^{ - 1}} \cr} $</p>
<p>so:</p>
<p>${{du} \over {dx}} = 1$
and ${{dv} \over {dx}} = - {v^{ - 2}}$
so:</p>
<p>$\eqalign{
& {{dy} \over {d... | DonAntonio | 31,254 | <p>Hint:</p>
<p>$$(uv^{-1})'=u'v^{-1}+u(v^{-1})'$$</p>
<p><strong>But</strong></p>
<p>$$(v^{-1})'=-\frac{v'}{v^2}\;,\;\;\text{so}\ldots$$</p>
|
2,640,909 | <p>I encountered a problem with 4 variables and I was wondering if anyone knows how to solve this:
This is what is known:</p>
<p>$$ \left\lbrace
\begin{align}
a+b &= 1800 \\
c+d &= 12 \\
a/c &= 100 \\
b/d &= 250 \\
(a+b)/(c+d) &= 150
\end{align}
\right.$$</p>
<p>Below is a screenshot from a spread... | Donald Splutterwit | 404,247 | <p>Linear algebra will do ... we have
\begin{eqnarray*}
a+b=1800 \\
c+d=12
\end{eqnarray*}
and $a=100c$, $b=250d$ sub these into the first equation & cancel a factor of $50$. (Note that the fifth equation is satisfied by virtue of the first two.)
\begin{eqnarray*}
2c+5d=36 \\
c+d=12
\end{eqnarray*}
Should be a dodd... |
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