qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | Bernard | 202,857 | <p><em>With a telescoping sum</em>: note that
<span class="math-container">$$(k+1)^4-k^4= 4k^3+6k^2+4k+1.$$</span>
Write this equation for <span class="math-container">$k=1, 2,\dots, n$</span> and add them. You'll need to know the sums <span class="math-container">$1+2+\dots +n$</span> and <span class="math-container... |
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | Robert Z | 299,698 | <p>Let <span class="math-container">$S_n^{(r)}=\sum_{k=1}^n k^r$</span>. Note that
<span class="math-container">$$(k+1)^4-k^4=4k^3+6k^2+4k+1$$</span>
and after summing for <span class="math-container">$k=1,\dots,n$</span>, (the sum on the left is telescopic) we get
<span class="math-container">$$(n+1)^4-1=4 S_n^{(3)}+6... |
2,867,907 | <p>Is the function </p>
<p>$$ f\left(\frac{x}{\epsilon}\right)=\exp\left(\frac{2\pi ix}{\epsilon}\right)=g(x) $$</p>
<p>equivalent to zero ? in the limit $ \epsilon \to 0 $ ?</p>
<p>If I take the derivative $$\frac{ g(x+\epsilon)-g(x)}{\epsilon} $$ is $0$ because the function $g(x)$ has a period 'epsilon'</p>
<p>Al... | Community | -1 | <p>You cannot take the derivative with the same $\epsilon$ as that in the function.</p>
<p>$$\lim_{\epsilon\to0}\frac{e^{2i\pi\epsilon/\epsilon}-1}\epsilon\ne
\lim_{\epsilon\to0}\lim_{\eta\to0}\frac{e^{2i\pi \eta/\epsilon}-1}\eta.$$</p>
|
3,649,221 | <blockquote>
<p>Suppose <span class="math-container">$\{f_n\}$</span> is an equicontinuous sequence of functions defined on <span class="math-container">$[0,1]$</span> and <span class="math-container">$\{f_n(r)\}$</span> converges <span class="math-container">$∀r ∈ \mathbb{Q} \cap [0, 1]$</span>. Prove that <span cla... | Danny Pak-Keung Chan | 374,270 | <p>Firstly, we show that <span class="math-container">$\lim_{n\rightarrow\infty}f_{n}(x)$</span> exists
for all <span class="math-container">$x\in[0,1]$</span>. Let <span class="math-container">$x\in[0,1]$</span>. Let <span class="math-container">$\varepsilon>0$</span> be arbitrary.
By equicontinuity, there exists <... |
3,484,293 | <p>In the <span class="math-container">$xy$</span> - plane, the point of intersection of two functions <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$g(x) = x + 2$</span> lies in which quadrant/s ?</p>
<p>I have no idea how to begin with this question.</p>
| The Demonix _ Hermit | 704,739 | <p><strong>Hint :</strong> We know that <span class="math-container">$f(x) = x^2 \ge 0$</span> for all <span class="math-container">$x$</span> . In which <em>quadrants</em> would this function lie ?</p>
<p>Also <span class="math-container">$f(x) = x+2$</span> is defined in all quadrants except <span class="math-contain... |
2,773,515 | <p>Given $X_1 \sim \exp(\lambda_1)$ and $X_2 \sim \exp(\lambda_2)$, and that they are independent, how can I calculate the probability density function of $X_1+X_2$? </p>
<hr>
<p>I tried to define $Z=X_1+X_2$ and then: $f_Z(z)=\int_{-\infty}^\infty f_{Z,X_1}(z,x) \, dx = \int_0^\infty f_{Z,X_1}(z,x) \, dx$.<br>
An... | drhab | 75,923 | <p>$$f_Z(z)=\int f_{X_1}(x)f_{X_2}(z-x)dx$$</p>
<p>Note that in your case the RHS has integrand $0$ if $z\leq0$ so that $f_Z(z)=0$ if $z\leq0$. </p>
<p>For $z>0$ we have:$$f_Z(z)=\int f_{X_1}(x)f_{X_2}(z-x)dx=\int_0^{z}f_{X_1}(x)f_{X_2}(z-x)dx$$</p>
<p>Work this out yourself.</p>
|
4,410,917 | <p>A student is looking for his teacher. There is a 4/5 chance that the teacher is in one of 8 rooms, and he has no specific room preferences. Student checked 7 of the rooms, but the teacher wasn't in any of them. What's the probability that he is in one of the 8 rooms?</p>
<p>I tried dividing the P(4/5) by 8 and getti... | drhab | 75,923 | <p><strong>Hint:</strong></p>
<p>Number the rooms with <span class="math-container">$i=1,2,\dots,8$</span>.</p>
<p>Let <span class="math-container">$E_i$</span> denote the event that the teacher is in room <span class="math-container">$i$</span>.</p>
<p>Now find:<span class="math-container">$$P(E_8|E_1^c\cap\cdots\cap ... |
753,553 | <p>Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre image of the map $L \to M$ and the image of the map $M \to N$ as they are submodules of $L$ and $N$ respectively, but I c... | jwsiegel | 143,003 | <p>Let $S_1\subset S_2 \subset S_3 \subset ... $ be an ascending chain of submodules
in $M$. </p>
<p>Now consider the chains $S_1\cap L \subset S_2\cap L \subset ... $
and $f(S_1)\subset f(S_2)\subset ... $ in $L$ and $N$, respectively
(here $f:M\rightarrow N$ is the second map in the exact sequence).</p>
<p>As $L$ a... |
2,378,577 | <p>How do I prove or disprove that for a rational number x and an irrational number y, $\ x^y\ $ is irrational?</p>
| Siong Thye Goh | 306,553 | <p>$2^{\log_2 3} =3$ is rational.</p>
<p>Check that $\log_2 3$ is irrational.</p>
<p>Suppose it is rational. </p>
<p>$$\log_2 3 = \frac{a}{b}$$ where $gcd(a,b)=1$.</p>
<p>$$3^b=2^a$$
which is a contradition.</p>
|
32,088 | <h2>Motivation</h2>
<p>One of the methods for strictly extending a theory <span class="math-container">$T$</span> (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of <span class="math-container">$T$</span> ( <span class="math-container">$Con(T)$</... | François G. Dorais | 2,000 | <p>The <a href="http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem" rel="nofollow">Paris–Harrington Theorem</a> is equivalent (over IΣ<sub>1</sub>) to Con(PA + Tr(Π<sub>1</sub>)), where Tr(Π<sub>1</sub>) is the set of all true Π<sub>1</sub> sentences in the language of arithmetic.</p>
<... |
1,791,631 | <p>The following is stated on <a href="https://en.wikipedia.org/wiki/Constructible_universe#L_and_large_cardinals" rel="nofollow">Wikipedia</a> for <a href="https://en.wikipedia.org/wiki/Mahlo_cardinal" rel="nofollow">Mahlo cardinals</a>. Unfortunately, it's not sourced. Where can I find details? I wasn't able to googl... | DanielWainfleet | 254,665 | <p>Let $C(k)$ be the set of club subsets of $k.$ Let $R(k)=\{l\in k: l=cf(l)\}.$ Observe that for any set $S\subset On,$ if $S\in L$ then </p>
<p>(1) $\forall a\in On \;[\;\{b\cap S :b\in a\} \in L\;], \; \text {and}$</p>
<p>(2) $\forall a\in On\;[\;a=\cup (a\cap S)\iff L\Vdash (a= \cup (a\cap S)\;].$</p>
<p>(3) A... |
1,676,347 | <p>Suppose that $V$ and $W$ are vector spaces, $g:V\rightarrow W$ a linear map. Show that g is surjective $\iff$ For any vector space $U$ the map $g^{\ast}:Hom(W,U) \rightarrow Hom(V,U)$ defined by $g^{\ast}(f) = f\circ g$ is injective.</p>
<p>I've failed too much trying to solve this problem. Any hint could be useful... | DonAntonio | 31,254 | <p>Suppose $\;g\;$ is surjective and assume $\;f\in\ker g^*\;,\;\;g^*:\text{Hom},(W,U)\to\text{Hom}\,(V,U)\;\implies$</p>
<p>$$g^*(f)=0\implies\color{red}{\forall v\in V}\;,\;\;f\circ g(v)=f(gv)=0\implies g(v)\in \ker f$$</p>
<p>But since $\;g\;$ is surjective, any $\;w\in W\;$ is $\;f(v_w)\;$ , for some $\;v_w\in V\... |
1,555,429 | <p>Hi I am trying to solve the sum of the series of this problem:</p>
<p>$$
11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots
$$</p>
<p>I know its a geometric series, but I cannot find the pattern around this. </p>
| Michael Hardy | 11,667 | <p><b>Q:</b> What does it mean to say a series is geometric?</p>
<p><b>A:</b> It means there is a common ratio.<br>${}\qquad{}$I.e. the number by which you multiply each term to get the next is the <b>same</b> in every case.</p>
<p>What do you multiply $11$ by to get $2$? You multiply it by $\dfrac 2 {11}$.</p>
<p>... |
2,505,757 | <p>Today, I was trying to prove <a href="https://math.stackexchange.com/questions/2505714/showing-cantor-set-is-uncountable">Cantor set is uncountable</a> and I completed it just a while ago.</p>
<p>So, I know that the end-points of each $A_n$ are elements of $C$ and those end-points are rational numbers. But since $C... | Eric Wofsey | 86,856 | <p>No Hausdorff space with more than one point is irreducible. Indeed, if $X$ is a Hausdorff space with two different points $x,y\in X$, there are disjoint open sets $U$ and $V$ with $x\in U$ and $y\in V$. The complements of $U$ and $V$ are then closed sets whose union is $X$ and neither is all of $X$, so $X$ is not ... |
956,256 | <p>If $a_n \ge 0 $ for all n, prove that $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty {a_n\over 1+a_n}$ converges. </p>
<p>Here is my attempt!</p>
<p>=> Suppose that $\epsilon \ge 0$ is given and $a_n$ converges, then for all $N\le n \le m$$$\sum_{k=n+1}^m a_k \lt \epsilon.$$
let $b_n={a_n\ove... | Leucippus | 148,155 | <p>Given $s(t) = 32 + 112 t - 16 t^{2}$ then $v(t)$, being the derivative of $s(t)$, is $v(t) = 112 - 32 t$. If $v(t) = 0$ then $t = 7/2$. Now, $s(7/2) = 32 + 56 \cdot 7 - 4 \cdot 49 = 228$. </p>
<p>To find the velocity at impact solve for $s(t) = 0$. This yields $16 t^{2} - 112 t - 32 = 0$, or $t^{2} - 7 t - 2 = 0$. ... |
3,715,987 | <p>The domain is: <span class="math-container">$\forall x \in \mathbb{R}\smallsetminus\{-1\}$</span></p>
<p>The range is: first we find the inverse of <span class="math-container">$f$</span>:
<span class="math-container">$$x=\frac{y+2}{y^2+2y+1} $$</span>
<span class="math-container">$$x\cdot(y+1)^2-1=y+2$$</span>
<sp... | user0102 | 322,814 | <p>Notice that
<span class="math-container">\begin{align*}
f(x) = \frac{x+2}{x^{2} + 2x + 1} = \frac{(x+1) + 1}{(x+1)^{2}} = \frac{1}{x+1} + \frac{1}{(x+1)^{2}}
\end{align*}</span></p>
<p>If we set <span class="math-container">$y = 1/(x+1)$</span> and consider <span class="math-container">$c\in\textbf{R}$</span>, we a... |
312,238 | <p>Reading my textbook, I came across exercises for nested quantifiers.</p>
<p>The question: Let $L(x, y)$ be the statement “$x$ loves $y$,” where the domain for both $x$ and $y$ consists of all people in the world.
Use quantifiers to express each of these statements.</p>
<p>i) Everyone loves himself or herself.</p>
... | Petr | 37,490 | <p>They're equivalent.</p>
<p>Let's start with $(\forall x)(\forall y) ((x=y)\to L(x,y))$:</p>
<ol>
<li>Apply <a href="https://en.wikipedia.org/wiki/Universal_instantiation" rel="noreferrer">instantiation</a> twice and set $x/x$ and $x/y$, getting $(x=x)\to L(x,x)$.</li>
<li>$x=x$ is an axiom of <a href="https://en.w... |
1,119,010 | <p>Write down the assumptions in a form of clauses and give a resolution proof that the proposition
$$\Big((p \rightarrow q) \land ( q \rightarrow r) \land p \Big) \rightarrow r$$
is a tautology.</p>
| Rob Arthan | 23,171 | <p>The negation of the goal has the following conjunctive normal form:
$$
\lnot(((p \rightarrow q) \land ( q \rightarrow r) \land p) \rightarrow r) \equiv
(\lnot p \lor q) \land (\lnot q \lor r) \land p \land \lnot r
$$
I.e., you have four clauses:
$$
\begin{array}{cl}
A:& \{\lnot p, q\}\\
B:& \{\lnot q, r\}\... |
3,971,833 | <p>If there is an <span class="math-container">$n$</span> by <span class="math-container">$n$</span> matrix where each element is either 1 or -1, how many unique matrices are there such that each row and each column multiplies to 1?</p>
<p>I solved for the trivial case of <span class="math-container">$n = 2$</span>, wh... | Calvin Lin | 54,563 | <p>Consider any <span class="math-container">$(n-1) \times (n-1) $</span> minor.<br />
This could be filled with any option of <span class="math-container">$ \pm 1$</span>.</p>
<p>Claim: The unfilled cells are uniquely determined by the column / row that they are in, and there is no conflict.</p>
<p>Proof: Do it yourse... |
3,459,205 | <p>Explain why <span class="math-container">$\arccos(t)=\arcsin(\sqrt{{1}-{t^2}})$</span> when <span class="math-container">$0<t≤1$</span>. </p>
<p>I tried researching online, couldn't find anything related to this question though. Know this equation is correct and make sense, just don't know how to explain it usin... | Semiclassical | 137,524 | <p>Here is a calculus proof: Given the integral definitions of <span class="math-container">$\arcsin$</span> and <span class="math-container">$\arcsin$</span>, which respectively are </p>
<p><span class="math-container">$$\arcsin t=\int_0^t \frac{dz}{\sqrt{1-z^2}},\\ \arccos t=\int_t^1 \frac{dz}{\sqrt{1-z^2}}$$</span>... |
3,622,508 | <p>I’m not sure exactly about the conditions needed for a subset <span class="math-container">$S$</span> to localise a ring <span class="math-container">$R$</span>. I know <span class="math-container">$S$</span> has to be multiplicative. But does <span class="math-container">$S$</span> also have to be a subset of the n... | James | 506,916 | <p>Localisation is defined for any commutative ring with identity <span class="math-container">$R$</span>, and for any multiplicatively closed subset <span class="math-container">$S \subset R$</span>: <span class="math-container">$1 \in S$</span> and <span class="math-container">$s,t \in S$</span> implies <span class="... |
484,313 | <p>I am taking linear algebra and none of this stuff is expained. I found this helpful link <a href="http://www.math.ucla.edu/~pskoufra/M115A-Notation.pdf" rel="nofollow">http://www.math.ucla.edu/~pskoufra/M115A-Notation.pdf</a></p>
<p>but it is missing a lot of what I need to know. Just right now though what does v a... | Adriano | 76,987 | <p><a href="http://en.wikipedia.org/wiki/List_of_logic_symbols" rel="nofollow">Wikipedia</a> has a list of logic symbols. The expression:</p>
<blockquote>
<p>$\forall x ~ [(x \in A \implies x \in B) ~~~\land~~~ (\exists x \in B ~\text{ s.t. }~ x \in A)]$</p>
</blockquote>
<p>can be interpreted to mean:</p>
<blockq... |
2,648,549 | <p>Let $\tau_{ij}$ be a transposition if degree n. What does it mean when one says that $\tau_{ij}=\tau_{ji}$? Thanks in advance!</p>
| projectilemotion | 323,432 | <p>It is possible that you have not learnt that the trace of a matrix is the sum of its eigenvalues and the determinant of a matrix is the product of its eigenvalues. Hence, I provide an alternative approach:</p>
<hr>
<p>Note that the following statements are equivalent:</p>
<ul>
<li>$\lambda$ is an eigenvalue of $P... |
2,796,694 | <p>So for my latest physics homework question, I had to derive an equation for the terminal velocity of a ball falling in some gravitational field assuming that the air resistance force was equal to some constant <em>c</em> multiplied by $v^2.$ <br> So first I started with the differntial equation: <br>
$\frac{dv}{dt}... | Community | -1 | <p>In order to determine the teriminal velocity, set $m\frac{dv}{dt}=-mg+cv^2=0$, which implies that $v_t=\sqrt{\frac{mg}{c}}$. The differential equation itself can be solved as follows. Since we know $v_t$, we can rewrite the orign differential equation as $\frac{dv}{dt}=g(1-\frac{v^2}{v_t^2})$ with boundary condition... |
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
4,008,987 | <p>I am reading a math book and in it, it says, "Let <span class="math-container">$V$</span> be the set of all functions <span class="math-container">$f: \mathbb{Z^n_2} \rightarrow \mathbb{R}.$</span> I know that <span class="math-container">$\mathbb{Z^n_2}$</span> is just the cyclic group of order <span class="ma... | mrtaurho | 537,079 | <p>A function is by definition a mapping between sets. So we have to look at the underlying sets. For simplicity, we write <span class="math-container">$\Bbb Z_2=\{0,1\}$</span>. Then <span class="math-container">$\Bbb Z_2^n=\{0,1\}^n$</span>. A map <span class="math-container">$f\colon\{0,1\}^n\to\Bbb R$</span> now is... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | ho boon suan | 118,745 | <p>Here's a chess problem. If one greedily places queens on an infinite <span class="math-container">$\mathbf{N}\times\mathbf{N}$</span> chessboard, column by column, such that at each step no two queens may attack one another, then one obtains a permutation of <span class="math-container">$\mathbf{N}$</span> (<a href=... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | László Kozma | 7,368 | <p>There are several exponential-time algorithms (in theoretical computer science, arguably a subfield of mathematics), whose running time can be bounded by an expression of the form <span class="math-container">${n-k \choose k}$</span> where <span class="math-container">$k$</span> is an integer between <span class="ma... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | coudy | 6,129 | <p>The golden L is a translation surface that has received much attention recently in the theory of billiard dynamics. It is built out of a <span class="math-container">$1 \times 1$</span> square with <span class="math-container">$1 \times \phi$</span> and <span class="math-container">$\phi\times1$</span> rectangles gl... |
403,184 | <p>A (non-mathematical) friend recently asked me the following question:</p>
<blockquote>
<p>Does the golden ratio play any role in contemporary mathematics?</p>
</blockquote>
<p>I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for al... | James Propp | 3,621 | <p>In the active field of research called symbolic dynamics (for a nice overview see <a href="https://www.southalabama.edu/mathstat/personal_pages/williams/wilshort.pdf" rel="nofollow noreferrer">https://www.southalabama.edu/mathstat/personal_pages/williams/wilshort.pdf</a> by Susan Williams), the canonical example of ... |
2,394,815 | <p>Let $A$ be a (not necessarily finitely generated) abelian group where all elements have order 1, 2, or 4. Does it follow that $A$ can be written as a direct sum $(\bigoplus _\alpha \mathbb Z/4) \oplus (\bigoplus_\beta \mathbb Z/2)$?</p>
| Eric Wofsey | 86,856 | <p>Yes. Note that such an abelian group is the same thing as a module over the ring <span class="math-container">$\mathbb{Z}/4$</span>. Note also that any direct sum of copies of <span class="math-container">$\mathbb{Z}/4$</span> is injective as a module over <span class="math-container">$\mathbb{Z}/4$</span> (this i... |
2,394,815 | <p>Let $A$ be a (not necessarily finitely generated) abelian group where all elements have order 1, 2, or 4. Does it follow that $A$ can be written as a direct sum $(\bigoplus _\alpha \mathbb Z/4) \oplus (\bigoplus_\beta \mathbb Z/2)$?</p>
| Steve D | 265,452 | <p>What I said in the comments is true, and goes by the name</p>
<p><strong>Prüfer's First Theorem</strong>: An abelian $p$-group $G$ with bounded exponent (an integer $k$ such that $g^k=1$ for all $g\in G$) is a direct sum of cyclic subgroups.</p>
<p>The proof is by induction on $k=p^e$, the base case $e=1$ being th... |
2,556,339 | <p>This is the function $f(x)$$=\frac{1}{\sqrt{3x-2}}$ .
I wrote that $$\lim_{h\to 0}\frac{\frac{\sqrt{3x+3h-2}}{3x+3h-2}-\frac{\sqrt{3x-2}}{3x-2}}{h}.$$
I am not able to continue further.</p>
| Jack | 510,857 | <p>Let $a = \sqrt{3x+h-2}$ and $b = \sqrt{3x-2}$, then $a^2 - b^2 = h$ which means $h \to 0$ as $a \to b$. So we've:</p>
<p>$$ \lim_{a \to b} \frac{1/a-1/b}{a^2-b^2} = \lim_{a \to b} \frac{b-a}{(ab)(a+b)(a-b)} = - \lim_{a \to b} \frac{1}{ab(a+b)} = -\frac{1}{2 b^3} = -\frac{1}{2\sqrt{(3x-2)^3}.} $$</p>
|
3,407,852 | <p>A space X is said to be h-homogeneous if
every non-empty clopen subset of <span class="math-container">$X$</span> is homeomorphic to <span class="math-container">$X.$</span></p>
<p>Is the space <span class="math-container">$L = 2^{\mathbb N} - \{p\}$</span> for <span class="math-container">$p \in 2^{\mathbb N}$</s... | Ross Millikan | 1,827 | <p>Compute the number of squares, cubes, and fifth powers. Now note that you have counted the sixth, tenth, and fifteenth powers twice each, so subtract them. You have counted the thirtieth powers three times and subtracted them three times, so add them once more.</p>
|
2,601,601 | <p>Consider the complex matrix $$A=\begin{pmatrix}i+1&2\\2&1\end{pmatrix}$$ and the linear map $$f:M(2,\mathbb{C})\to M(2,\mathbb{C}),\qquad X\mapsto XA-AX.$$</p>
<p>I want to find a basis of $\ker f$.</p>
<p>I already know the canonical basis $\{E_{11},E_{12},E_{21},E_{22}\}$ and computed $$f(E_{11})=\begin{... | Cameron Buie | 28,900 | <p>It does. It means that for an arbitrary matrix $$X=\begin{pmatrix}a&b\\c&d\end{pmatrix},$$ we have $$f(X)=af(E_{11})+bf(E_{12})+cf(E_{21})+df(E_{22}),$$ or $$f(X)=(a-d)f(E_{11})+(b-c)f(E_{12}).$$ Thus, we have $f(X)$ is the zero matrix if and only if...what?</p>
|
3,523,213 | <p>I've read some simple explanations of Cantor's diagonal method.</p>
<p>It seems to be:</p>
<pre><code>1) Changing the i-th value in a row.
2) Do the same to the next row with the (i+1)th element.
3) Now you get an element not in any other row. So add it to list.
4) This process never ends.
</code></pre>
<p>This l... | Vsotvep | 176,025 | <p>Cantor's diagonal proof is <strong>not</strong> infinite in nature, and neither is a proof by induction an infinite proof. </p>
<hr>
<p>For Cantor's diagonal proof (I'll assume the variant where we show the set of reals between <span class="math-container">$0$</span> and <span class="math-container">$1$</span> is ... |
2,332,419 | <p>What's the angle between the two pointers of the clock when time is 15:15? The answer I heard was 7.5 and i really cannot understand it. Can someone help? Is it true, and why?</p>
| Steven Alexis Gregory | 75,410 | <p>Lets assume we have a 12 hour clock and let 1200 hrs be 0 degrees and lets measure angles clockwise from there. At 1500 hrs, the hour hand is exactly on the three, that is at 90 degrees.</p>
<p>It takes the hour hand 12 hours to travel 360 degrees. That comes to
$\dfrac{360^\circ}{12 \text{hrs}} = 30 \frac{\text{d... |
3,745,551 | <p>I often see people say that if you have 2 IID gaussian RVs, say <span class="math-container">$X \sim \mathcal{N}(\mu_x, \sigma_x^2)$</span> and <span class="math-container">$Y \sim \mathcal{N}(\mu_y, \sigma_y^2)$</span>, then the distribution of their sum is <span class="math-container">$\mathcal{N}(\mu_x + \mu_y, \... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent , <span class="math-container">$X \sim N(\mu_X,\sigma_X^{2})$</span> and <span class="math-container">$Y \sim N(\mu_Y,\sigma_Y^{2})$</span> then <span class="math-container">$X+Y \sim N(\mu_X+\mu_Y,\sigma_X^{2}+\s... |
253,271 | <p>I recently found that <code>\[Gradient]</code> and <code>\[InlinePart]</code> both expand (contract) to special symbols in MMA.</p>
<p>So far as I can tell (see <a href="https://mathematica.stackexchange.com/questions/134506/inlinepart-what-is-it-and-what-happened-to-it">InlinePart. What is it and what happened to i... | Syed | 81,355 | <p>The list of keyboard shortcuts for the Wolfram language and System is <a href="https://reference.wolfram.com/language/tutorial/KeyboardShortcutListing.html" rel="nofollow noreferrer">here</a>.</p>
<p>The list of named characters for the Wolfram language and System is <a href="https://reference.wolfram.com/language/g... |
393,467 | <p>I am looking for a proof that:</p>
<p>if <span class="math-container">$A_{11}A_{12}...A_{1n}$</span>; <span class="math-container">$A_{21}A_{22}...A_{2n}$</span>; <span class="math-container">$\cdots$</span>; <span class="math-container">$A_{i1}A_{i2}...A_{in}$</span>; <span class="math-container">$\cdots$</span>; <... | Algernon | 23,297 | <p>To complement <a href="https://mathoverflow.net/a/393477/23297">Iiro Ullin</a>'s answer, we have the following inequality in one direction:</p>
<p><strong>Lemma</strong> (<a href="https://doi.org/10.1109/TIT.2005.864431" rel="nofollow noreferrer" title="I. Csiszar and Z. Talata,Context tree estimation for not necess... |
393,467 | <p>I am looking for a proof that:</p>
<p>if <span class="math-container">$A_{11}A_{12}...A_{1n}$</span>; <span class="math-container">$A_{21}A_{22}...A_{2n}$</span>; <span class="math-container">$\cdots$</span>; <span class="math-container">$A_{i1}A_{i2}...A_{in}$</span>; <span class="math-container">$\cdots$</span>; <... | Mark | 101,207 | <p>This is probably obvious, but just wanted to explicitly mention the following.
It is perhaps more natural to look at the modified <span class="math-container">$L_2$</span> norm <span class="math-container">$L_2'(p,q) = L_2(\sqrt{p}, \sqrt{q})$</span>. Note that probability distributions (which are a subset of unit v... |
1,252,167 | <p>I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. </p>
<p>It seems that functions can be manipulated as vectors as long as they are not interpreted as having real values.<br>
Suppose the solution space of a linear homogeneous di... | triple_sec | 87,778 | <p>I'm not sure if it helps you with your original question, but let me try proposing an alternative concept capturing the intuitive notion of irredundancy: for each point, take the “smallest” open set in the cover containing that point.</p>
<p>Formally, let $\{U_{\lambda}\}_{\lambda\in\Lambda}$ be an open cover of th... |
1,824,966 | <p>Ok, I was asked this strange question that I can't seem to grasp the concept of..</p>
<blockquote>
<p>Let $T$ be a linear transformation such that:
$$T \langle1,-1\rangle = \langle 0,3\rangle \\
T \langle2, 3\rangle = \langle 5,1\rangle $$
Find $T$.</p>
</blockquote>
<p>Is there suppose to be a funct... | Mnifldz | 210,719 | <p>Yes, $T$ is a matrix of some kind. You can tell that it is $2\times 2$ since both its inputs and outputs are vectors of length $2$. My recommendation for solving this is the following. Suppose $T$ has matrix representation</p>
<p>$$
T \;\; =\;\; \left [ \begin{array}{cc}
a & b \\
c & d \\
\end{array} \ri... |
3,663,526 | <p><span class="math-container">$F(x)=\begin{cases} x^3+5, & x\ge 1\\ x^3+2, & 0\leq x<1\\ x^3, & x<0 \end{cases}$</span></p>
<p>Let <span class="math-container">$\mu_{F}$</span> be the Lebesgue-Stieltjes measure associated with <span class="math-container">$F$</span>. Find the lebesgue decomposition... | Ng Tze Beng | 866,955 | <p>F is an increasing function, so you have the Lebesgue decomposition for the positive Lebesgue Stieltjes measure. See for example Theorem 8 of <em>Complex Measure, Dual Space of L
p Space, Radon-Nikodym Theorem and Riesz Representation Theorems</em>. (<a href="https://my-calculus-web.firebaseapp.com/MA3110/Complex%20... |
85,717 | <p>Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$.</p>
<p>There are many different categories that one can associate to a space $X$. For example, one could build the small category whose object set is the set of points wi... | Peter May | 14,447 | <p>Could I ask young people to use precise language? Calling a Kan complex an $\infty$-groupoid
and asking what kind of category it is just jars. It feels so pointless (I'm toning down the language I'm tempted to use). As Tony pointed out, a topological category in the proposed sense is just a category internal to t... |
174,573 | <p>I am generating a random matrix $F$ and then plotting norm of the matrices $E+Ft$, representing them as a table. But I also want to print the matrix $F$ near every plot. If I remove the semicolon from <code>F // MatrixForm;</code> in the code below I get some errors.</p>
<p>My code:</p>
<pre><code>Dimension := 3;
... | kglr | 125 | <p><strong>Update:</strong> <code>f4</code> and <code>f5</code> should work as is in version 8. Variants of <code>f1</code>, <code>f2</code> and <code>f3</code> that avoid the functions and forms that are only available in later versions are: </p>
<pre><code>ClearAll[f1b, f2b, f3b]
f1b[l_List, {p__}] := Boole[MatchQ[#... |
3,928,937 | <p>Determine all the solutions of the congruence<br />
<span class="math-container">$x^{85} ≡ 25 \pmod{31}$</span><br />
using index function in base <span class="math-container">$3$</span> module <span class="math-container">$31$</span>.<br />
It is clear to me that <span class="math-container">$3$</span> is primitive... | J. W. Tanner | 615,567 | <p><span class="math-container">$3^22^2\equiv5 $</span> and <span class="math-container">$2^2\equiv(-3)^3$</span>, so <span class="math-container">$3^5\equiv -5$</span>, and <span class="math-container">$3^{10}\equiv25\pmod{31}$</span>.</p>
<p>Let <span class="math-container">$x=3^y$</span>, so you're asking for <span ... |
3,815,990 | <p>The following question was asked in End Term exam of real analysis and I was clueless on how it can be approached .</p>
<p>So, I am asking for guidence here.</p>
<blockquote>
<p>Question: Let <span class="math-container">$\phi :[0,1] \to \mathbb{R}$</span> be a continuous function such that <span class="math-contain... | TheSilverDoe | 594,484 | <p><strong>Hint :</strong> Show, using the appropriate version of DCT, that the function
<span class="math-container">$$F : a \mapsto \int_0^1 \phi(t)e^{-at} dt$$</span></p>
<p>is <span class="math-container">$\mathcal{C}^{\infty}$</span>, and that for all <span class="math-container">$n \in \mathbb{N}$</span>,
<span c... |
3,815,990 | <p>The following question was asked in End Term exam of real analysis and I was clueless on how it can be approached .</p>
<p>So, I am asking for guidence here.</p>
<blockquote>
<p>Question: Let <span class="math-container">$\phi :[0,1] \to \mathbb{R}$</span> be a continuous function such that <span class="math-contain... | zhw. | 228,045 | <p>If you are familiar with the Stone-Weierstrass theorem: Let <span class="math-container">$\mathcal A$</span> be the algebra of functions of the form <span class="math-container">$\sum_{k=1}^{n}a_ke^{-kt}.$</span> By S-W, <span class="math-container">$\mathcal A$</span> is dense in <span class="math-container">$C[0,1... |
787,558 | <p>we have $ad-bc >1$ is it true that at least one of $a,b,c,d$ is not divisible by $ad-bc$ ?
Thanks in advance.</p>
<p><strong>Example:</strong>
$a=2$ , $b = 1$, $c = 2$, $d = 2$, $ad-bc = 2$ </p>
<p>so $b$ is not divisible by $ad-bc$</p>
| Magdiragdag | 35,584 | <p>Yes, that is true. </p>
<p><strong>Hint.</strong> Consider a common divisor $q$ of $a, b, c, d$. What does that mean for $q^2$ and $ad - bc$? Now assume that $ad - bc$ is a common divisor of $a, b, c, d$ and apply this to $ad - bc$ itself. This is going to give a contradiction with the assumption that $ad - bc >... |
379,194 | <p>Let $X$ be a topological space and $X^*$ be its supspace. It is stated in my textbook that if $c(A)$ represents the closure of set $A$ in $X$, then $c(A) \bigcap X^*$ is closed in $X^*$. </p>
<p>A closed set is one which contains all its limit points, and a limit point of a set is a point such that every open set c... | Jared | 65,034 | <p>Since the problem is a power series, we must first write it in summation notation, and then differentiate term by term within its radius of convergence. Note we can write the power series as
$$\sum_{n=0}^{\infty}\frac{(x-1)^{2n+1}}{8n+4}$$
Now, we apply the ratio test
$$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\ri... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Timothy Chow | 3,106 | <p>The convergence conditions for the Fourier series of a function <span class="math-container">$f:S^1 \to \mathbb{R}$</span> are a good example. The investigation of convergence conditions for Fourier series was a major motivation for Cantor's set theory and Lebesgue's measure theory. Depending on what kind of converg... |
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Timothy Chow | 3,106 | <p><a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof" rel="noreferrer">Gödel's ontological proof</a> requires a subtle assumption that if <span class="math-container">$\varphi$</span> is an <i>essential property</i> of <span class="math-container">$x$</span> then <span class="math-container">$x$</... |
4,651,047 | <p>I've recently come across an interesting integral, which is of the form:</p>
<p><span class="math-container">$$\int_0^1\arctan(x)\log\left(\frac{1-x}{1+x}\right)\mathrm{d}x$$</span></p>
<p>To start, I expanded the arctangent into its series expansion, then utilized the Weierstraß substitution in order to remove the ... | Quanto | 686,284 | <p>Integrate by parts with <span class="math-container">$dx=d(x-1)$</span></p>
<p><span class="math-container">\begin{align}
&\int_0^1\tan^{-1}x\ln\frac{1-x}{1+x}\ {dx}\\
\overset{ibp}= & \int_0^1 \frac{(1-x)\ln \frac{1-x}{1+x}}{1+x^2}\ \overset{\frac{1-x}{1+x}\to x}{dx}
-2\int_0^1 \frac{\tan^{-1}x}{1+x}dx\\
=&... |
4,651,047 | <p>I've recently come across an interesting integral, which is of the form:</p>
<p><span class="math-container">$$\int_0^1\arctan(x)\log\left(\frac{1-x}{1+x}\right)\mathrm{d}x$$</span></p>
<p>To start, I expanded the arctangent into its series expansion, then utilized the Weierstraß substitution in order to remove the ... | xpaul | 66,420 | <p>Noting
<span class="math-container">$$ d\bigg[(x-1)\ln\bigg(\frac{1-x}{1+x}\bigg)-2\ln(1+x)\bigg]=\ln\bigg(\frac{1-x}{1+x}\bigg) $$</span>
one has, by IBP,
<span class="math-container">\begin{eqnarray}
&&\int_0^1\tan^{-1}x\ln\bigg(\frac{1-x}{1+x}\bigg)\ {dx}\\
&=&-\int_0^1\bigg[(x-1)\ln\bigg(\frac{1-... |
1,041,177 | <p>Proof that if $p$ is a prime odd and $k$ is a integer such that $1≤k≤p-1$ , then the binomial coefficient</p>
<p>$$\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod p$$</p>
<p>This exercise was on a test and I could not do!!</p>
| erfan soheil | 195,909 | <p>You can prove it by induction on $k$.
If $ k=1$ $\to$ $\displaystyle \binom{p-1}{k} = p-1$ that $p-1 \equiv -1 \mod p$.
For $k= n +1$ use this
$\displaystyle \binom{m}{n+1} =\displaystyle \binom{m}{n} +\displaystyle \binom{m}{n+1}$</p>
|
2,311,979 | <p>Let $A = (a_{i,j})_{n\times n}$ and $B = (b_{i,j})_{n\times n}$</p>
<p>$(AB) = (c_{i,j})_{n\times n}$, where $c_{i,j} = \sum_{k=1}^n a_{i,k} b_{k,j}$, so</p>
<p>$(AB)^T = (c_{j,i})$, where $c_{j,i} = \sum_{k=1}^n a_{j,k}b_{k,i} $, and
$B^T = b_{j,i}$ and $A^T = a_{j,i}$, so </p>
<p>$B^T A^T = d_{j,i}$ where $d_... | 5xum | 112,884 | <p>OK, your notations confusing. It's very hard to understand what you mean by $(c_{j,i})$, so let's start over.</p>
<hr>
<p>Let's call $AB = C$, and let's call $B^TA^T=D$. What we want to prove is $C^T=D$.</p>
<p>First of all, let's say that the element in $C$'s $i$-th row and $j$-th column is $c_{ij}$. Then you kn... |
1,463,881 | <blockquote>
<p>By considering $\sum_{r=1}^n z^{2r-1}$ where z= $\cos\theta + i\sin\theta$, show that if $\sin\theta$ $\neq$ 0, $$\sum_{r=1}^n \sin(2r-1)\theta=\frac{\sin^2n\theta}{\sin\theta}$$</p>
</blockquote>
<p>I couldn't solve this at first but with some hints some of you gave, I was able to come up with my ow... | Jack D'Aurizio | 44,121 | <p>The number of solutions of
$$ x_1+x_2+x_3+x_4+x_5= 5 $$
with the constraints $x_1\in[0,3],x_2\in[0,2]$ and $x_3,x_4,x_5\geq 0$ is given by the coefficient of $z^5$ in the product:
$$ (1+z+z^2+z^3)(1+z+z^2)(1+z+z^2+\ldots)(1+z+z^2+\ldots)(1+z+z^2+\ldots)$$
i.e. by the coefficient of $z^5$ in the Taylor series around... |
4,330,755 | <p>Given a convex pentagon <span class="math-container">$ABCDE$</span>, there is a unique ellipse with center <span class="math-container">$F$</span> that can be inscribed in it as shown in the image below. I've written a small program to find this ellipse, and had to numerically (i.e. by iterations) solve five quadra... | achille hui | 59,379 | <p><strong>Disclaimer</strong> - this is not an independent answer!</p>
<blockquote>
<p>I have transformed the geometric construction in Intelligenti pauca's excellent answer into algebra using homogeneous coordinates. From that,
I observed the center of inellipse can be expressed as a weighted sum of vertices (<span ... |
149,049 | <p>Suppose you have a list of intervals (or tuples), such as:</p>
<pre><code>intervals = {{3,7}, {17,43}, {64,70}};
</code></pre>
<p>And you wanted to know the intervals of all numbers not included above, e.g.:</p>
<pre><code>myRange = 100;
numbersNotUed[myRange,intervales]
(*out: {{1,2},{8,16},{44,63},{71,100}}*)... | kglr | 125 | <pre><code>f1 = Partition[Flatten[{0, Select[Flatten@#, Function[x, x < #2]], #2 + 1}],
2, 2, {1, -1}, {}, ({1, -1} + {##}) /. {1, 0} -> Nothing &]&;
f1[{{3, 7}, {17, 43}, {64, 70}}, 80]
</code></pre>
<blockquote>
<p>{{1, 2}, {8, 16}, {44, 63}, {71, 80}}</p>
</blockquote>
<p>Also</p>
<pre><co... |
86,202 | <p>Let $\mathcal{L},\mathcal{U}$ be invertible sheaves over a
noetherian scheme $X$, where $X$ is of finite type over a noetherian
ring $A$. If $\mathcal{L}$ is very ample, and $\mathcal{U}$ is
generated by global sections, then $\mathcal{L} \otimes \mathcal{U}$
is very ample.</p>
<p>Since $\mathcal{L}$ is very ample,... | KReiser | 21,412 | <p>From the comments, to clear this from the unanswered queue: </p>
<blockquote>
<p>Let <span class="math-container">$\pi:\Bbb P^m\to \operatorname{Spec} A$</span> be the projection. Then, <span class="math-container">$(\text{id}\times\pi)\circ (i\times j) \circ \Delta = i$</span> is an immersion, while <span class=... |
844,700 | <p>I am looking for a calculator which can calculate functions like $f(x) = x+2$
at $x=a$ etc; but I am unable to do so. Can you recommend any online calculator?</p>
| murkle | 177,867 | <p>GeoGebra works well for this, eg
<a href="http://web.geogebra.org/?command=f(x)=x%2B2;a=2;f(a)" rel="nofollow">http://web.geogebra.org/?command=f(x)=x+2;a=2;f(a)</a></p>
|
2,623,560 | <blockquote>
<p>Decide if $\mathbb Z[i]/\langle i\rangle$ and $\mathbb Z$ are isomorphic, if $\mathbb Z[i]/\langle i+1\rangle$ and $\mathbb Z_2$ are isomorphic</p>
</blockquote>
<p>I know that in the first case if there exist such homomorphism then $f(i)=0$ (and in the second case $f(i+1)=0$), but I don't know exa... | egreg | 62,967 | <p>It seems you are in the context of rings.</p>
<p>Note that $i$ is invertible in $\mathbb{Z}[i]$, so the ideal it generates is the whole ring. Hence the quotient $\mathbb{Z}[i]/\langle i\rangle$ is the trivial ring.</p>
<p>For the second part, consider $a+bi=a-b+b(1+i)$ and the ring homomorphism
$$
\varphi\colon\ma... |
834,949 | <p>I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$.
And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$.
Am I right?</p>
| André Nicolas | 6,312 | <p>To get an <em>expression</em> for the $n$-th derivative, you can use the <a href="http://en.wikipedia.org/wiki/General_Leibniz_rule" rel="nofollow">Leibniz Rule</a>
$$(fg)^{(n)}=\sum_{k=0}^n \binom{n}{k}f^{(k)}g^{(n-k)}.$$</p>
<p>Here $h^{(i)}$ denotes the $i$-th derivative of $h$. </p>
|
1,554 | <p>Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?</p>
<p>(Penny Smith and I wrote a paper on this many years ago, but we... | Anton Petrunin | 1,441 | <p>Once we considered a similar problem but around infinity,
try to look in our paper "Asymptotical flatness and cone structure at infinity".</p>
<p>Let us denote by $r$ the distance to the singular point.
If dimensions $\not= 4$ then the same method shows that at singular point we have Euclidean tangent cone even if ... |
4,369,232 | <p>I have the following problem:</p>
<blockquote>
<p>Let <span class="math-container">$\{(M_i,\tau_i)\}_{i\in I}$</span> be nonempty topological spaces where <span class="math-container">$I$</span> is arbitrary but non empty. Let <span class="math-container">$M=\prod_{i\in I} M_i$</span>. Let <span class="math-containe... | Nicolás Vilches | 413,494 | <p>You have to be a bit careful with the details here. When you write <span class="math-container">$U=\prod_{i \in I} O_i$</span>, it is not true that every open set of <span class="math-container">$M$</span> is of this form. Rather, you can show that given a neighborhood <span class="math-container">$p \in U$</span>, ... |
456,583 | <p>I was searching for a Latex symbol that indicates $A \Rightarrow B$ and $A \not\Leftarrow B$ ($B$ if not only if $A$, $B$ ifnf $A$). I thought of using $A \Leftrightarrow B$ with the left arrow tick <code><</code> crossed out. Since I did not find such a symbol:</p>
<p>Is there a Latex symbol for this?</p>
<p>H... | Emily | 31,475 | <p>It is doubtful that a symbol exists; I do not believe it is common usage.</p>
<p>Note that your situation is equivalent to "$A$ implies $B$, but $B$ does not imply $A$". There are many, many situations in mathematics when this is the case. For instance:</p>
<blockquote>
<p>Independent random variables have zero ... |
34,487 | <p>A few years ago Lance Fortnow listed his favorite theorems in complexity theory:
<a href="http://blog.computationalcomplexity.org/2005/12/favorite-theorems-first-decade-recap.html" rel="nofollow">(1965-1974)</a>
<a href="http://blog.computationalcomplexity.org/2006/12/favorite-theorems-second-decade-recap.html" rel=... | Marcos Villagra | 7,692 | <p>I think you should add as a recent result the proof for QIP=IP=PSPACE</p>
|
704,921 | <p>This is the question:
$$
\frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2
$$
I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some guidance? Usually I turn all the bases to 2, and then work with the powers, but I probaby make the same mistake ever... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use $$a^{mn}=(a^m)^n$$ and $$a^m\cdot a^n=a^{m+n}\text{ and }\frac{a^m}{a^n}=a^{m-n}$$</p>
<p>For example, $\displaystyle4^{8+n}=(2^2)^{(8+n)}=2^{2(8+n)}$</p>
<p>Finally $a^x=a^y\implies x=y$ if real $a\ne\pm1$ </p>
<p><strong>Reference</strong> : <a href="http://www.proofwiki.org/wiki/Exponent_Com... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | Igor Rivin | 11,142 | <p>The simplest version of this question is: what is the quickest way to evaluate $x^n?$ For $n = 2^k,$ $k$ repeated squarings is obviously best, but for more complicated $n$ I believe that finding the optimum is very hard -- see Knuth, vol 2 for (much) more on these so-called "multiplication trees".</p>
|
823,928 | <p>Prove that all of the rings, which mediate between principal ideal ring $K$ and the field of fractions $Q$, are the principal ideal ring.</p>
| John Machacek | 155,418 | <p>$K = \mathbb{Z}$</p>
<p>$R = \{\frac{a}{2^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$</p>
<p>$Q = \mathbb{Q}$</p>
<p>If I understand your question here is an counter example.</p>
|
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Olumide | 7,486 | <p>I believe I now have the answer to the question. The power of $\omega$ appear from the taylor expansion of $e^{i\omega.t_j}$ (in section 2.3 of Kent and Mardia's paper)</p>
<p>Thanks.</p>
<p>(Apologies for the seeming bit of self promotion, but I've tagged this as the correct answer.)</p>
|
2,280,243 | <blockquote>
<p>A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its <em>seeds</em> For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $... | Doug M | 317,176 | <p>I would consider the matrix <span class="math-container">$T = \begin{bmatrix} 1&1&1\\1&0&0\\0&1&0\end{bmatrix}$</span>.</p>
<p>And note that when applied to any sequence <span class="math-container">$$S = \begin{bmatrix} s_3\\s_2\\s_1\end{bmatrix}$$</span></p>
<p><span class="math-container">... |
2,927,079 | <p><a href="https://i.stack.imgur.com/ih7X2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ih7X2.png" alt="enter image description here"></a></p>
<p>In the second paragraph, Munkres assumes that there exists a separation of <span class="math-container">$Y$</span> (in the sense he defined in Lemma 2... | Steve Kass | 60,500 | <p>I agree that this is unclear. It looks to me as though Munkres means “Suppose <span class="math-container">$Y$</span> is not connected.”</p>
<p>The first use of “separation” in the proof seems to be his way to say that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> form a witnes... |
1,651,427 | <blockquote>
<p>Let $f$ be a bounded function on $[0,1]$. Assume that for any $x\in[0,1)$, $f(x+)$ exists. Define $g(x)=f(x+)$, $x\in [0,1)$, and $g(1)=f(1)$. Is $g(x)$ right continuous? </p>
</blockquote>
<p>Prove it or give me a counterexample.</p>
<p>My ideas:</p>
<p>$(1)$If $f$ is of bounded variation, then $g... | Qiyu Wen | 310,935 | <p>Given any $\epsilon > 0$ and any $x \in [0,1)$, pick $\delta > 0$ such that $|f(y) - f(x+)| < \epsilon/2$ for all $y \in (x,x+\delta)$. For any $y \in (x,x+\delta)$, let $\{x_n\}$ and $\{y_n\}$ be two sequences approaching $x$ and $y$ respectively from the right. Then there exists an integer $N$ such that $... |
1,651,427 | <blockquote>
<p>Let $f$ be a bounded function on $[0,1]$. Assume that for any $x\in[0,1)$, $f(x+)$ exists. Define $g(x)=f(x+)$, $x\in [0,1)$, and $g(1)=f(1)$. Is $g(x)$ right continuous? </p>
</blockquote>
<p>Prove it or give me a counterexample.</p>
<p>My ideas:</p>
<p>$(1)$If $f$ is of bounded variation, then $g... | Phillip Hamilton | 312,810 | <p>We are assuming that $f(x+)$ exists. Its definition is:</p>
<p>$f(x+) = q$ where $q\in [0,1)$, if $f(t_n) \to q$ as $n \to \infty$ for all {$t_n$} in $[x,1)$ s.t. $t_n \to x$</p>
<p>So try setting $g(x) = f(x+)$, then think about {$t_n$}</p>
|
194,218 | <blockquote>
<p>Let A, B be two sets. Prove that <span class="math-container">$A \subset B \iff A \cup B = B$</span></p>
</blockquote>
<p>I'm thinking of using disjunctive syllogism by showing that <span class="math-container">$\neg \forall Y(Y \in A).$</span> However, I'm not sure how the proving steps should proceed ... | Community | -1 | <p>Let $A\subset B$. Since $B\subset B$, we have $A\cup B\subset B$. Clearly, $B\subset A\cup B$. Hence $A\cup B=B$.</p>
<p>Let $A\not\subset B$. Then there is some $x\in A$ with $x\not\in B$. Clearly, $x\in A\cup B$. Hence $A\cup B\neq B$.</p>
|
194,218 | <blockquote>
<p>Let A, B be two sets. Prove that <span class="math-container">$A \subset B \iff A \cup B = B$</span></p>
</blockquote>
<p>I'm thinking of using disjunctive syllogism by showing that <span class="math-container">$\neg \forall Y(Y \in A).$</span> However, I'm not sure how the proving steps should proceed ... | Iuli | 33,954 | <p>For the first implication a draw can help us:
<img src="https://i.stack.imgur.com/ySAJW.png" alt="enter image description here"></p>
<p>and now it is obvious. </p>
<p>Now conversely, we have: </p>
<p>$A \cup B =B \Rightarrow$ $A \cup B \subset B \tag{1}$ and $B \subset A \cup B\tag{2}.$ We need only the relation... |
194,218 | <blockquote>
<p>Let A, B be two sets. Prove that <span class="math-container">$A \subset B \iff A \cup B = B$</span></p>
</blockquote>
<p>I'm thinking of using disjunctive syllogism by showing that <span class="math-container">$\neg \forall Y(Y \in A).$</span> However, I'm not sure how the proving steps should proceed ... | Asaf Karagila | 622 | <p>Taking the "longer" road. Let us review the definitions:</p>
<ol>
<li>$A\subseteq B$ if and only if <em>for every $x\in A$, $x\in B$</em>.</li>
<li>$x\in A\cup B$ if and only if $x\in A$ <strong>or</strong> $x\in B$.</li>
<li>$A=B$ if and only if $A\subseteq B$ <strong>and</strong> $B\subseteq A$.</li>
<li>$P\iff Q... |
194,218 | <blockquote>
<p>Let A, B be two sets. Prove that <span class="math-container">$A \subset B \iff A \cup B = B$</span></p>
</blockquote>
<p>I'm thinking of using disjunctive syllogism by showing that <span class="math-container">$\neg \forall Y(Y \in A).$</span> However, I'm not sure how the proving steps should proceed ... | A.P. Phaneendra kumar | 811,292 | <p>Let <span class="math-container">$A\subseteq B$</span>.<br>
<strong>Claim:</strong> We need to prove <span class="math-container">$A\cup B\subseteq B$</span> and <span class="math-container">$B\subseteq A\cup B$</span></p>
<p>Let <span class="math-container">$x\in A\cup B$</span>. Then <span class="math-container... |
113,963 | <p>While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by E. Kolchin, nor the texts by Ph. J. Cassidy contain anything like this — they work only with the groups of points ov... | Peter | 69,733 | <p>For difference algebraic groups, I think the paper</p>
<p>Michael Wibmer: <a href="http://arxiv.org/abs/1405.6603" rel="nofollow">Affine difference algebraic groups</a></p>
<p>is what you asked for.</p>
|
150,180 | <p>I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:</p>
<p>Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross u... | Qiaochu Yuan | 290 | <blockquote>
<p>(obviously, in the affine case this question translates into: can the category of (finitely generated) modules be defined via the category of projective modules (of finite rank)?)</p>
</blockquote>
<p>Yes (you're assuming Noetherian here, right?). We will need to combine two observations. Let $A$ be ... |
3,599,893 | <p>I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a very large sphere. (Stretching or shrinking of different places is OK... | James K | 92,207 | <p>Although you can't make a sphere from a plane, there are map projections that tessellate "naturally" (and place the tricky singular points in the ocean where people tend not to notice them). You can't, for topological reasons, avoid the points at the corners, but this kind of map does avoid some of the problems of m... |
2,468,067 | <p>Can we say that that if $f(x)$ and $f^{-1}(x)$ intersect, then at least one point of intersection will lie on $y=x$? </p>
<p>Also there are many function e.g. $f(x)=1-x^3$ where point of intersection exists outside $y=x$ There will be $5$(odd) point of intersection of $f(x)=1-x^3$ and $f^{-1}(x)=(1-x)^{1/3}$ ou... | hmakholm left over Monica | 14,366 | <p>if the graphs of $f$ and $f^{-1}$ intersect at $(x,y)$, then they will also intersect at $(y,x)$.</p>
<p>If the function is additionally assumed to be <em>continuous</em>, then -- since $(x,y)$ and $(y,x)$ are on different sides of the diagonal $x=y$ -- the function graph must cross the line $x=y$. And such a point... |
3,891,749 | <p>I got a pretty good idea for the proof but it feels like it's missing some details.</p>
<p>Proof:
<span class="math-container">$$X \times Y \Leftrightarrow a\in X \land b \in Y \Leftrightarrow a\in X \land b\in Z$$</span></p>
<p>Since <span class="math-container">$b\in Y \Leftrightarrow b\in Z$</span>, then <span cl... | Hagen von Eitzen | 39,174 | <p>You do not clearly state where and how you use <span class="math-container">$X\ne \emptyset$</span>, and you should start from <span class="math-container">$Y$</span> or <span class="math-container">$Z$</span> and not the products.</p>
<p>Let <span class="math-container">$y\in Y$</span> be arbitrary. As <span class=... |
1,032,650 | <p><img src="https://i.stack.imgur.com/GVk1i.png" alt="enter image description here"></p>
<p>Here, ABCD is a rectangle, and BC = 3 cm. An Equilateral triangle XYZ is inscribed inside the rectangle as shown in the figure where YE = 2 cm. YE is perpendicular to DC. Calculate the length of the side of the equilateral tri... | Jack D'Aurizio | 44,121 | <p>Consider the reference system with the origin in $Z$ in which $DC$ is the real axis, and let $EZ=a$. Then we have $Y=a+2i$ and:</p>
<p>$$e^{\pi i/6}(a+2i) = X $$
so:
$$\Im \left[\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\cdot\left(a+2i\right)\right]=3, $$
or:
$$ 1+\frac{\sqrt{3}}{2}a = 3 $$
so $a = \frac{4}{\sqrt... |
3,242,363 | <blockquote>
<p>Why does this function, <span class="math-container">$$\tan\left(x ^ {1/x}\right)$$</span>
have a maximum value at <span class="math-container">$x=e$</span>?</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/pqE0Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pqE0Q.png" ... | Acccumulation | 476,070 | <p>The general term for this is "simultaneous equations". In this case, as Ovi notes, they are linear equations. You can write is as</p>
<p><span class="math-container">$$1a+1b+-y=-x$$</span>
<span class="math-container">$$1a-.029b=.029x+.3$$</span>
<span class="math-container">$$.015a-1b=-.015x$$</span></p>
<p>In ma... |
554,025 | <p>I have a question as such:</p>
<blockquote>
<p>Class A has 45 students in it, and class B has 30 students in it. In class A, every student attends any particular lecture with probability 0.7 independent of the other students. For class B, two thirds of lectures are attended by everyone, with probability 1/3 that a s... | N. S. | 9,176 | <p>Pick any five vertices. Then, as $K_5$ is not planar, there is a pair of edges in the graph generated by these 5 vertices which cross. </p>
<p>Thus, we get at least one crossing for every combination of $5$ vertices. In total, our count is at least $\binom{n}{5}$.</p>
<p>But, we might had counted the same crossing... |
2,828,487 | <p>If $\mathcal{R}$ is a von Neumann algebra acting on Hilbert space $H$, and $v \in H$ is a cyclical and separating vector for $\mathcal{R}$ (hence also for its commutant $\mathcal{R}'$), and $P \in \mathcal{R}, Q \in \mathcal{R}'$ are nonzero projections, can we have $PQv = 0$?</p>
<p>[note i had briefly edited this... | José Carlos Santos | 446,262 | <p>As far as textbooks are concerned, I learned Real Analysis mainly from Michael Spivak's <em>Calculus</em>. My contact with Walter Rudin's textbooks started only after graduation, but I also have a very high opinion about them.</p>
|
2,828,487 | <p>If $\mathcal{R}$ is a von Neumann algebra acting on Hilbert space $H$, and $v \in H$ is a cyclical and separating vector for $\mathcal{R}$ (hence also for its commutant $\mathcal{R}'$), and $P \in \mathcal{R}, Q \in \mathcal{R}'$ are nonzero projections, can we have $PQv = 0$?</p>
<p>[note i had briefly edited this... | Thomas | 26,188 | <p><strong>People will have different experiences of doing research.</strong> </p>
<p>Typically in a graduate program in the US one starts with taking general classes. After covering some basic classes (often in algebra, topology, and analysis (and maybe others)) you might start to take more specialized classes. At th... |
3,070,258 | <p>In a (Partial Differential Equations / Laplace Equation) , I try to solving a problem of Laplace eq. by using separation of variables method.</p>
<p>I usually using the rule : if <span class="math-container">$e^{2 \sqrt{k} b} = 1$</span>, then I have: <span class="math-container">$2\sqrt{k} b = 2ni\pi$</span>. </p>... | Rhys Hughes | 487,658 | <p>You're using Euler's formula for complex numbers</p>
<p><span class="math-container">$$e^{i\theta}=\cos(\theta)+i\sin(\theta)$$</span></p>
<p>We know that <span class="math-container">$\cos(2n\pi)=1, \sin(2n\pi)=0$</span>, for <span class="math-container">$n\in \Bbb Z$</span> so:</p>
<p><span class="math-containe... |
930,611 | <blockquote>
<p>Find the maximal value of the function for $a=24.3$, $b=41.5$:
$$f(x,y)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$$</p>
</blockquote>
<p>Using the second derivative test for partial derivatives, I find the critical point in terms of $a$ and $b$ by taking partial derivatives of $x$ and $y$ and equa... | Macavity | 58,320 | <p>Another way is to consider equivalently the maximum of
$$\frac{f^2}{a^2b^2}=\frac{x^2}{a^2} \frac{y^2}{b^2} \left(1-\frac{x^2}{a^2} -\frac{y^2}{b^2} \right)$$
which is the product of three positive terms with constant sum, so each term must be equal to (in this case) $\frac13$ at maximum.</p>
|
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| Bernard | 202,857 | <p>Set $u_n=n^{(-1)^n}$? Explicitly, $u_{2n}=2n$, $u_{2n+1}=\dfrac1{2n+1}$.</p>
<p>If the sequence were convergent, all subsequences would converge to the same limit. However we see the subsequence of odd terms converges to $0$, whereas the subsequence of even terms tends to $+\infty$.</p>
|
2,011,181 | <blockquote>
<p><strong>Question:</strong> Find the area of the shaded region given $EB=2,CD=3,BC=10$ and $\angle EBC=\angle BCD=90^{\circ}$.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/BFf2h.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BFf2h.jpg" alt="Diagram"></a></p>
<p>I first ... | kotomord | 382,886 | <p>Coordinates:</p>
<p>B (0,0)</p>
<p>E(0,2)</p>
<p>C (10, 0)</p>
<p>D( 10, 3)</p>
<p>BD 10y - 3x = 0</p>
<p>EC x+5y = 10</p>
<p>Find y coordinate of A:
10y - 3x = 0, 3x+15y = 30 => 25y = 30 => y = 6/5</p>
<p>Area size is 10*(6/5)/2 = 6;</p>
|
451,063 | <p>Alright so I am having the following issue: I want to figure out how to find the fourier coefficients of the following function:
$$D(X)=\frac {a'(x)} {1+a'(x)^2}$$</p>
<p>Where $a(x)$ is an arbitrary function. I already have a model for finding the fourier coefficients for $a(x)$ and $a'(x)$:</p>
<pre><code>fc =... | AnonSubmitter85 | 33,383 | <p>The DFT is isomorphic, so if you have the DFT coefficients for $a'$, then all you need to do to get $a'$ is apply the inverse DFT:</p>
<pre><code>D = ifft(aprimec) ./ (1 + ifft(aprimec).^2);
</code></pre>
|
4,378,215 | <p>Is there are a big-O notation for a function <span class="math-container">$f(x)$</span> that tends <span class="math-container">$\infty$</span> at an arbitrarily small rate?</p>
<p>Obviously, the expression <span class="math-container">$\mathcal{O(1)}<f(x)<\mathcal{O}(\sqrt{\log(x)})$</span> is not good enough... | Thomas Lesgourgues | 601,841 | <p>You want to use the notation <span class="math-container">$f(x)=\omega(1)$</span>.</p>
<p>The notation <span class="math-container">$f = \omega(g)$</span>, <span class="math-container">$f$</span> dominate <span class="math-container">$g$</span>, is defined as
<span class="math-container">$$\forall k > 0 \, \exist... |
4,378,215 | <p>Is there are a big-O notation for a function <span class="math-container">$f(x)$</span> that tends <span class="math-container">$\infty$</span> at an arbitrarily small rate?</p>
<p>Obviously, the expression <span class="math-container">$\mathcal{O(1)}<f(x)<\mathcal{O}(\sqrt{\log(x)})$</span> is not good enough... | Arthur | 15,500 | <p>There isn't a good way to do this with big-<span class="math-container">$O$</span> notation. No matter what expression you put into it, you can take a logarithm or a square root to find a significantly smaller function that still goes to infinity.</p>
<p>But there are other (albeit less common) notations for similar... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Steve Huntsman | 1,847 | <p>Practically, mathematicians today should know the rudiments of programming in at least one language (Mathematica and MATLAB count). They should know the basics of probability and linear algebra. They should know these three things because if they get jobs outside of academia they will generally be expected to use at... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Joel David Hamkins | 1,946 | <p>As mathematics grows and diversifies beyond belief, surely the collection of topics that <em>every</em> mathematician must know is shrinking fast. One can carry out serious mathematical research in one area while knowing very little of another, even when many mathematicians regard that other area as fundamentally im... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | Kevin O'Bryant | 935 | <p>One thing I'm sure we'll <em>all</em> agree on: every mathematician should know some flavor of TeX!</p>
|
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