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 |
|---|---|---|---|---|
1,293,773 | <p>Suppose the function $f :\mathbb{R}\to\mathbb{R}$ has limit $L$ at $0$, and let $a > 0$. If $g :\mathbb{R}\to\mathbb{R}$ is defined by
$g(x) := f(ax)$ for $x\in \mathbb{R}$. Show that $\displaystyle{\lim_{x\to 0} g(x) = L}$.</p>
| MCT | 92,774 | <p>Let $\varepsilon > 0$. We wish to show that there is some $\delta$ such that $|f(ax) - L| < \varepsilon$ whenever $|x| < \delta$.</p>
<p>Well, we know that there is some $\hat \delta$ such that $|f(x) - L| < \varepsilon$ whenever $|x| < \hat \delta$, so if we take $\delta = \frac{\hat \delta}{a}$ th... |
4,388,965 | <p>Let <span class="math-container">$h(X) = X/2$</span>, and <span class="math-container">$f(X) = (3X + 1)/2$</span>. Then clearly every iteration <span class="math-container">$g^i(X), X \in \Bbb{Z}$</span> the Collatz mapping <span class="math-container">$$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \... | JetfiRex | 1,014,890 | <p>To be honest, we can prove it by induction.</p>
<p>Suppose for a <span class="math-container">$m$</span>-length sequence <span class="math-container">$f,h$</span> there exists a modulo <span class="math-container">$M$</span> such that for any starting number <span class="math-container">$k=2^mt+M$</span>, the first ... |
4,590,386 | <p>I was reviewing some books of logic and find that the statement "(∃!x)P(x)" is equivalent to "(∃x)P(x) ∧ (∀y)(∀z)(P(y) ∧ P(z) ⇒ y = z)", i understand that "(∃x)P(x)" is read as "there exist a x that, P(x)", therefore "(∀y)(∀z)(P(y) ∧ P(z) ⇒ y = z)" is the part that m... | Bram28 | 256,001 | <p>The statement reads: For any objects <span class="math-container">$y$</span> and <span class="math-container">$z$</span>: if both <span class="math-container">$y$</span> and <span class="math-container">$z$</span> have property <span class="math-container">$P$</span>, then <span class="math-container">$y$</span> an... |
2,136,757 | <p>Let $H=\{ x \in \Bbb R : x+\frac3x-1 \le 0\}$. Find $\sup(H)$, $\max(H)$, $\inf(H)$, and $\min(H)$. I understand the terminology but I am new to this please just offer any first steps or any help on how to get started.</p>
| Martin Sleziak | 8,297 | <p>Notice that we have
$$x+\frac3x \ge 2\sqrt3$$
for $x>0$. There are various ways to show this, you can have a look at several posts with proofs of
<a href="https://math.stackexchange.com/q/705064">similar inequality $x + \frac{1}{x} \geq 2 $</a>.</p>
<p>Let us denote $f(x)=x+\frac3x-1$. For $x>0$ you have
$$... |
3,891,910 | <p>Let <span class="math-container">$U$</span> be a n by n matrix, satisfy <span class="math-container">$U^T=-U$</span></p>
<p>Prove that <span class="math-container">$U-I$</span> is invertible.</p>
<p>So, I have some hints, <span class="math-container">$\vec{v}^T\vec{v}$</span> for <span class="math-container">$\vec{v... | Darsen | 637,522 | <p>If <span class="math-container">$v$</span> is a vector such that <span class="math-container">$(U-I)v=0$</span> then <span class="math-container">$Uv=Iv=v$</span>, and taking the transpose <span class="math-container">$v^\mathsf tU^\mathsf t=-v^tU=v^t$</span>. Multiplying by <span class="math-container">$v$</span> o... |
3,891,910 | <p>Let <span class="math-container">$U$</span> be a n by n matrix, satisfy <span class="math-container">$U^T=-U$</span></p>
<p>Prove that <span class="math-container">$U-I$</span> is invertible.</p>
<p>So, I have some hints, <span class="math-container">$\vec{v}^T\vec{v}$</span> for <span class="math-container">$\vec{v... | Robert Lewis | 67,071 | <p>I assume <span class="math-container">$U$</span> is a real matrix.</p>
<p>Note that</p>
<p><span class="math-container">$(I + U)(I - U) = I - U^2; \tag 1$</span></p>
<p>given</p>
<p><span class="math-container">$U^T = -U, \tag 2$</span></p>
<p>we have</p>
<p><span class="math-container">$U^2 = -U^T U = -UU^T, \tag 3... |
2,451,634 | <p>$$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$ I tried somthing,like go to the power of two , and change of variables... but it became more complicated . Is there an idea to solve this system of equation ?
Thanks in advance </p>
| Andreas | 317,854 | <p>So here are the 3 equations:</p>
<p>$$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$</p>
<p>As suggested by transcenmental,</p>
<p>$\sqrt{x-1}-\sqrt{y-1}=4\sqrt 3 - 2 \sqrt{y-1}$ and multiplying with the first eq. gives </p>
<p>$$
x - y = 4\... |
2,451,634 | <p>$$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$ I tried somthing,like go to the power of two , and change of variables... but it became more complicated . Is there an idea to solve this system of equation ?
Thanks in advance </p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p>Eliminate $y$ and $z$,</p>
<p>$$y=(4\sqrt3-\sqrt{x-1})^2+1,\\z=(4\sqrt3-\sqrt{x-9})^2+9$$</p>
<p>and </p>
<p>$$\sqrt{44+x-8\sqrt3\sqrt{x-1}}+\sqrt{44+x-8\sqrt3\sqrt{x-9}}=4\sqrt3.$$</p>
<p>By plotting, you can see that this equation has two real solutions. It is possible, by succes... |
1,109,139 | <p>Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function
$$\theta:U\to\mathbb{R}$$
such that
$$e^{i\theta(z)}=z$$
for all $z\in U$ if and only if $U\neq S^1$?</p>
<p>I have been able to show the $\Rightarrow$ direction but not the other one... | Tim Raczkowski | 192,581 | <p>Let $\theta (z)=\Im (\log z)$ where $\log z$ is the branch for which $0<\arg z<2\pi$. This should work unless I'm missing something.</p>
|
3,935,057 | <p>Since A is countable and B is uncountable then this would be undefined as B is larger than A right? Or am I missing something here?</p>
| Y. Tamer | 633,238 | <p>Let <span class="math-container">$A=\{-1\}$</span>, <span class="math-container">$B=\mathbb{R}^+$</span>. Clearly, <span class="math-container">$A$</span> is countable and <span class="math-container">$B$</span> is uncountable. However, <span class="math-container">$A \backslash B=A$</span> is countable.</p>
<p>In f... |
2,239,956 | <p>As we can always hear that topology is related to graph theory. But I can only read some conceptual relations between them.
Technically, is it possible to write down a topology $\mathscr{T}_V$ (collection of open sets) of any given graph $V$ to consider the graph as a topological space?</p>
| Hagen von Eitzen | 39,174 | <p>Sure, there are many ways.
The following might be particularly useful:
Let the underlying set be $V\cup E$, and let $U\subseteq V\cup E$ be open iff for all $v\in U\cap V$, all edges incident with $v$ are also $\in U$.</p>
|
18,931 | <blockquote>
<p>(Application of the rank-nullity theorem) Suppose $S,T: V\to V$ are linear transformations of a finite dimensional vector space $V$, and that the composition $ST\colon V\to V$ is invertible. Show that then $T$ is one-to-one, and deduce that it is invertible. Show also that $S$ is invertible. Deduce th... | Pierre-Yves Gaillard | 660 | <p>There is a much stronger result which is not really harder to prove:</p>
<blockquote>
<p>Let $K$ be a commutative ring, let $V$ be a finitely generated $K$-module, let $T$ be a surjective $K$-linear endomorphism of $V$. Then $T$ is injective.</p>
</blockquote>
<p>Indeed, let $(v_j)$ be a finite system generating... |
2,203,876 | <p>As part of a proof regarding reduced homology, I need to know that the following fact holds for good pairs $(X,A)$:</p>
<p>If $q:X\rightarrow X/A$ is the quotient map, then $q|_{X\setminus A}:X\setminus A \rightarrow (X/A)\setminus (A/A)$ is a homeomorphism.
I can see why the restriction is injective, surjective (... | Niels J. Diepeveen | 3,457 | <p>The following appears to be a minimal counterexample.</p>
<p>Let $X = \{ 1,2,3,4 \}$ with the topology $\{ \emptyset, \{ 3,4 \}, X \}$ and let $A = \{ 2,3 \}$. It is clear
that $\{4\}$ is open in $X\setminus A$. On the other hand
$X/A$ has the trivial topology, since $\{3,4\}$ is not a union of equivalence classes.... |
2,379,932 | <p>Since the the powerset is defined as $\mathcal {P}(A)=\{x\mid x\subseteq A\} $, it seems obvious that $A \subseteq \mathcal P(A)$. But I don't know how to show it formally.<br>
Is it okay to say that $x\subseteq A$ (definition of a powerset), thus $A\subseteq \mathcal P(A)$?</p>
<p>It seems trivial, yet I don't kno... | lhf | 589 | <p>$A \in \mathcal {P}(A)$ but in general $A \not\subseteq \mathcal {P}(A)$ because the <em>elements</em> of $A$ are not necessarily <em>subsets</em> of $A$.</p>
<p>Note the difference between $\in$ (is an element of) and $\subseteq$ (is a subset of).</p>
|
1,278 | <p>Being quite new to the site, I'm curious if there is any established protocol with regards to action toward answers that are incorrect (e.g., produce incorrect / incomplete / etc. results). Is it appropriate to comment on the answer as such, or to place such a caveat on one's own answer if provided for the question?... | Artes | 184 | <p>To do is to be - Descartes<br>
To be is to do - Voltaire<br>
Do be do be do be do - Frank Sinatra<br>
You should do it <a href="http://www.youtube.com/watch?v=egY8rUpxqcE&list=RDegY8rUpxqcE" rel="nofollow"><strong>Your Way</strong></a>. </p>
|
1,410,498 | <p>How would I prove this using mathematical induction:
$\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$.</p>
<p>Here is what I have tried:</p>
<p>$\dfrac{n^n}{3^n} < n!$ for all $n \geq 6$</p>
<p>Base case: $\dfrac{6^6}{3^6} < 6!$. Yes, this is true.</p>
<p>Assume for $k$ it's true so $\dfrac{k^k}{3^k} < k!$... | Dr. Sonnhard Graubner | 175,066 | <p>we have to show that $$(n+1)!>\frac{(n+1)^{n+1}}{3^{n+1}}$$ multiplying $$n!>\frac{n^n}{3^n}$$ by $$n+1$$ we get
$$(n+1)!>\frac{n^n(n+1)}{3^n}$$ this must be greater as $$\frac{(n+1)^{n+1}}{3^{n+1}}$$ this is true since $$\left(1+\frac{1}{n}\right)^n<3$$</p>
|
2,481,743 | <p>Squaring both sides of $\sqrt{x} - \sqrt{2-2x} = 1$ and rearranging
I arrive at the quadratic $9x^2 - 10x + 1 = 0$ which has solutions $x=1/9$ and $x=1$.
I don't understand why $x=1$ fits the original equation but $x=1/9$ doesn't (left hand side gives $-1$).</p>
| orlp | 5,558 | <p>$$\sqrt{x} - 1 = \sqrt{2-2x}$$
$$x -2\sqrt{x} + 1 = 2-2x$$
$$3x -2\sqrt{x} - 1 = 0$$</p>
<p>Now if $\sqrt{x} = u$ we have:
$$3u^2 -2u - 1 = 0$$
$$u = 1 \vee u=-\frac{1}{3}$$</p>
<p>In the case $u = 1$ we've found the solution $x = 1$. But in the case $u = -\dfrac{1}{3}$ we get:</p>
<p>$$\sqrt{x} = -\frac{1}{3}$$<... |
2,481,743 | <p>Squaring both sides of $\sqrt{x} - \sqrt{2-2x} = 1$ and rearranging
I arrive at the quadratic $9x^2 - 10x + 1 = 0$ which has solutions $x=1/9$ and $x=1$.
I don't understand why $x=1$ fits the original equation but $x=1/9$ doesn't (left hand side gives $-1$).</p>
| ℋolo | 471,959 | <p>Just like dividing by $f(x)$ loose you the solution $x_0$ where $f(x_0)=0$ Multiply by $f(x)$ can add you solutions, so you need to check if the solutions you got are actually working.</p>
|
2,328,775 | <p>let $f(x,y)$ be a homogeneous function of order 2. What is the order of
$-x\cdot (f(x,y))^{2}$?</p>
<p>More generally: </p>
<blockquote>
<p><strong>Q.</strong> How do you determine the order of the product of two homogeneous functions, and what happens when you multiply a homogeneous function by a variable such... | hamam_Abdallah | 369,188 | <p>The characteristic equation is
$$x^2-2x+2=0.$$
the roots are</p>
<p>$1+i$ and $1-i $.</p>
<p>they are different from $1.i $, thus</p>
<p>the particular solution will be
$$y_p=A\sin (x)+B\cos (x) $$</p>
|
2,328,775 | <p>let $f(x,y)$ be a homogeneous function of order 2. What is the order of
$-x\cdot (f(x,y))^{2}$?</p>
<p>More generally: </p>
<blockquote>
<p><strong>Q.</strong> How do you determine the order of the product of two homogeneous functions, and what happens when you multiply a homogeneous function by a variable such... | Community | -1 | <p>The solution of the associated characteristic equation is </p>
<p>$$r_{1,2} = 1 \pm \mathrm{i}$$</p>
<p>for which we note that the product of the roots $r_1 r_2 = 2$ and the sum $r_1 + r_2 = 2$. Therefore, the complete solution to your ODE looks like</p>
<p>$$ y(x) = \mathrm{e}^{x} (A \cos{x} + B \sin{x} ) + y_p(... |
2,328,775 | <p>let $f(x,y)$ be a homogeneous function of order 2. What is the order of
$-x\cdot (f(x,y))^{2}$?</p>
<p>More generally: </p>
<blockquote>
<p><strong>Q.</strong> How do you determine the order of the product of two homogeneous functions, and what happens when you multiply a homogeneous function by a variable such... | k.Vijay | 428,609 | <p><strong>By Inverse Operator Rule:</strong> When $f(D)y=X$ where X is a function of $x$ or constant then,</p>
<p>Particular Integral, $P.I.=\dfrac{1}{f(D)}X$, Where $D=\dfrac{d}{dx}$.</p>
<p>Now when $X=\sin(ax+b)$, or, $\cos(ax+b)$ then
\begin{align*}
P.I.&=\dfrac{1}{f(D^2)}\sin(ax+b)\ \text{ put }D^2=-a^2,\t... |
3,429,075 | <blockquote>
<p>Evaluate <span class="math-container">$\lim\limits_{x\to 0} \cot ^2 (x)-\dfrac{1}{x^2}.$</span> </p>
</blockquote>
<p>I was thinking of using L'Hôpital's Rule, but things got very, very ugly so I wasn't able to solve it. I know from experimentation that the limit is <span class="math-container">$-\df... | J.G. | 56,861 | <p>Since <span class="math-container">$\sin x=x(1-\frac16x^2+o(x^2))$</span> and <span class="math-container">$\cos x=1-\frac12x^2+o(x^2)$</span>,<span class="math-container">$$\cot^2x=\cos^2x\sin^{-2}x=\frac{1}{x^2}\left(1+2\left(\frac16-\frac12\right)x^2+o(x^2)\right)=\frac{1}{x^2}-\frac23+o(1).$$</span></p>
|
1,212,672 | <p>I wish to prove that $\lim_{x\to 2} {x+1 \over x+2} = {3 \over 4} $.</p>
<p>The $(\delta,\varepsilon)$ limit definition in this case is:</p>
<p>$\forall \epsilon >0, \exists \delta >0$ such that $0<|x-2|<\delta \Rightarrow |{x+1 \over x+2} - {3 \over 4}| < \epsilon.$</p>
<p>Thus, I need to provide ... | JMP | 210,189 | <p>Using what you have written:</p>
<p>$$|{x+1 \over x+2} - {3 \over 4}| = {1 \over 4}|{x-2 \over x+2}|$$</p>
<p>as $x \to 2$, from either above or below, $x-2\to0$ and therefore is certainly $\lt \epsilon$.</p>
<p>If we write $2+\delta$ instead of $x$ we get: $${1 \over 4}|{2+\delta-2 \over 2+\delta+2}|={1 \over 4}... |
1,212,672 | <p>I wish to prove that $\lim_{x\to 2} {x+1 \over x+2} = {3 \over 4} $.</p>
<p>The $(\delta,\varepsilon)$ limit definition in this case is:</p>
<p>$\forall \epsilon >0, \exists \delta >0$ such that $0<|x-2|<\delta \Rightarrow |{x+1 \over x+2} - {3 \over 4}| < \epsilon.$</p>
<p>Thus, I need to provide ... | Lundborg | 227,259 | <p>You need to prove that $\forall \varepsilon > 0$ $\exists \delta > 0$ : $|x-2|<\delta \implies |\dfrac{x+1}{x+2}-\dfrac{3}{4}|<\varepsilon$</p>
<p>You've gotten a long way now that you have that $|\dfrac{x+1}{x+2}-\dfrac{3}{4}|=\dfrac{1}{4}|\dfrac{x-2}{x+2}|$.</p>
<p>What you want is an expression of t... |
788,344 | <p>I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. </p>
<p>But I'd also like to show the following 2 things:</p>
<p>1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = \Gamma(1)=1$</p>
<p>The second equality is obvious, but how about the limit equality?</p>
<p>2) Using 1) ... | Disintegrating By Parts | 112,478 | <p>The limit follows from the functional property of the Gamma function:
$$
a\Gamma(a)=\Gamma(a+1).
$$</p>
|
905,298 | <p>How I can calculate the integers between $20000$ and $30000$ that end in $39$, and end in $33$ in base $4$, and end in $37$ in base $8$.</p>
<p>I think that I have to solve the system of congruences:</p>
<p>$\qquad 4^x \equiv 33 \pmod{100}$</p>
<p>$\qquad8^x \equiv 37 \pmod{100}$</p>
<p>$\qquad\ x \equiv 39 \pmo... | Bill Dubuque | 242 | <p>$x\,$ ends in $\,\color{#c00}3\color{#0a0}7\,$ in base $8\iff x\, =\, n\cdot 8^2\! + \color{#c00}3\cdot 8 + \color{#0a0}7\iff x\equiv 3\cdot 8+7\equiv 31\pmod{8^2}$</p>
<p>$x\,$ ends in $\,\color{#c00}3\color{#0a0}3\,$ in base $4\iff x\, =\, n\cdot 4^2\! + \color{#c00}3\cdot 4 + \color{#0a0}3\iff x\equiv 3\cdot 4... |
12,394 | <p>There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.</p>
<p>Suppose $X$ is a set equipped with a metric $d$. I had initially assumed there must be an $n$ such that $X$ embeds isometrically into $\mathbb{R}^n$, but t... | Anton Petrunin | 1,441 | <p>There is a related question <a href="https://mathoverflow.net/questions/7794/feasibility-of-a-list-of-prescribed-distances-in-r3/7799#7799">here</a> (with an answer similar to Andrew's --- it answers question #1).</p>
<p>Question #2a: NO, say take 4 points p,x,y,z such that |px|=|py|=|pz|=1 and <span class="math-con... |
1,842,689 | <p>I hold a bachelor's degree in mathematics and I have taken undergraduate real analysis course. But I'm interested to know more about it, especially the stuff that will help me to understand more of topology. Could you please recommend me some books that I can start with?</p>
<p>Your help will be appreciated</p>
<p... | Dave L. Renfro | 13,130 | <p>Carothers' book that Bungo mentioned seems like it would be a good fit for you, but given what you wrote I think the following might be an even better fit:</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/1575242389" rel="nofollow"><strong>Introduction to Topology and Modern Analysis</strong></a> by George ... |
848,575 | <p>I have been doing GRE practice and saw following reasoning on one of the question</p>
<blockquote>
<p>When x and y intercepts are of same sign, the intercept would be negative</p>
</blockquote>
<p>Now this is confusing me</p>
<p>slope can be expressed as</p>
<p>y = mx + b</p>
<p>assuming b = 0 here</p>
<p>y ... | ngenisis | 160,159 | <p>Do you mean that if the intercepts are the same sign, then the slope is negative? Clearly you can have a line where both intercepts are positive; e.g. $x+y=1$. </p>
<p>If so, then suppose the line has $x$-intercept $(a,0)$ and $y$-intercept $(0,b)$ where $a$ and $b$ are either both positive or both negative. Then t... |
1,067,382 | <p>Is there any way to find the <code>n-th</code> number in the series, by knowing it's genereting function.</p>
<p>For example, I found that the closed form solution for a generating function $\displaystyle\frac{(1+x)^2}{(1-x)^3}$ can be expressed as $2n^2 + 2n + 1$.</p>
<p>I am interested how can I derive similar f... | Claude Leibovici | 82,404 | <p>My approach has been extremely empirical; so, please, forgive me for the lack of rigor !</p>
<p>What I first did was to develop as Taylor series (at $x=0$) to get $$\frac{(1+x)^5}{(1-x)^6}=1+11 x+61 x^2+231 x^3+681 x^4+1683 x^5+3653 x^6+7183 x^7+13073 x^8+22363 x^9+36365
x^{10}+O\left(x^{11}\right)$$ in which th... |
1,027,281 | <p>$1^x = 1^y$, and $x,y \in \mathbb{R}$.</p>
<p>Following the rule, same base has powers equal every $x$ should be equal to every $y$.
$$1^x = 1^y$$
$$x = y$$
What went wrong?</p>
| daOnlyBG | 173,397 | <p>$1^x=1^y$ does not imply $x=y$:</p>
<p>$$x\cdot \ln(1)=y\cdot\ln(1)$$
$$x \cdot 0=y\cdot 0$$</p>
<p>Note that $x,y$ can be anything- including different values, i.e., $x ≠y.$</p>
<p>The rule you mention is incorrect.</p>
|
4,224 | <p>Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?</p>
<p>I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers wi... | Jonas Meyer | 1,119 | <p>If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of the summands. This would be true more generally for commuting normal matrices. For arbitrary positive matrices, the largest eigenvalue of the sum will be less than or equal to the sum of the largest eigenvalues of the summan... |
1,026,647 | <p>I'm trying to integrate the following:</p>
<p>$\int_{0}^{\infty} x^2 e^{-2x}$</p>
<p>I know that $\int_{0}^{\infty} x^n e^{-x} dx = n!$</p>
<p>Feels like a $u$ substitution problem but I'm having trouble making use of the above.</p>
<p>Thanks,
Mariogs</p>
| bjd2385 | 167,604 | <p>Look at the more general integral, related to Laplace transforms, of this type:
\begin{align*}
\mathcal{L}\left\{t^n\right\}\left(s\right)&=\int_0^\infty t^n e^{-st}\:dt,
\end{align*}
and you may solve using integration by parts to always get something of the form
\begin{align*}
\int_0^\infty t^n e^{-st}\:dt &am... |
1,187,894 | <p><img src="https://i.stack.imgur.com/LFtIh.jpg" alt="enter image description here"></p>
<p>I thought I knew how to solve this kind of problems, but I really don't know how to start...</p>
| Alexander | 176,662 | <p>Integral Transform is a huge subject. In my opinion, you should also have a strong background in Ordinary Differential Equation, Partial Differential Equation, and Real/Complex Analysis. Linear Algebra and Calculus is a "must" known subjects if you want to know where does Integral Transform come from. On the other h... |
1,412,428 | <blockquote>
<p>Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $ \mathbb Z$-module for $m < n$ ?</p>
</blockquote>
<p>I think it's not possible. It might be a easy problem for some of you, but I really don't have any clean way to show this. Any ideas?</p>
| user26857 | 121,097 | <p>If $0\to\mathbb Z^n\to\mathbb Z^m$ is an exact sequence of $\mathbb Z$-modules, then $0\to S^{-1}\mathbb Z^n\to S^{-1}\mathbb Z^m$ is an exact sequence of $S^{-1}\mathbb Z$-modules, where $S=\mathbb Z-\{0\}$, hence $0\to\mathbb Q^n\to\mathbb Q^m$ is an exact sequence of $\mathbb Q$-vector spaces, so $n\le m$.</p>
|
1,585,908 | <p>I am currently self-studying complex analysis in preparation for my course that will commence next year and I am stuck with this question as shown below:</p>
<p>In the complex plane, draw a picture of $$S=\{z \in \mathbb{C} : \vert z-1 \vert + \vert z+1 \vert = 2\}.$$</p>
<p>I was told that by squaring both sides,... | Narasimham | 95,860 | <p>Sum of distances is 2. And inter-focal distance $ 2 a $ also 2. So it is a degenerate ellipse (or parabola to a scale). If the latter is > 2 an ellipse can be formed with foci at $( \pm 1, 0 )$</p>
|
581,944 | <p>I stuck at part (a) after 1 have done the partial fraction, can anyone give me some hints?
<img src="https://i.stack.imgur.com/UAcjO.png" alt=""></p>
<p>for part (b) I express $$f_1(z)=(-1)^n-1(n-1)!f_n(z)$$
What I suppose to do next? </p>
| John Machacek | 155,418 | <p><a href="http://math.gmu.edu/~lawrence/m689/l4.pdf" rel="nofollow">http://math.gmu.edu/~lawrence/m689/l4.pdf</a></p>
<p>This link has the steps to proving that all rank 2 oriented matroids are realizable. It leaves many details as exercises, but walks through the basic outline of the proof.</p>
|
202,012 | <p>Lets say I have a two dimensional list:</p>
<pre><code>list = {{a, b, c}, {aa, bb, cc}, {x, y, z}, {xx, yy, zz}};
</code></pre>
<p>and two pure functions:</p>
<pre><code>f1 = Total[#]/Length[#] &;
f2 = Total[##]/Length[##] &;
</code></pre>
<p>I want to learn how to apply these functions over rows and col... | Henrik Schumacher | 38,178 | <p>One of the most efficient ways (for machine precision data in a packed array) would be</p>
<pre><code>Total[list, {1}]/Length[list]
</code></pre>
<blockquote>
<p>{1/4 (a + aa + x + xx), 1/4 (b + bb + y + yy), 1/4 (c + cc + z + zz)}</p>
</blockquote>
<p>What is special here is that <code>Total</code> has a two-a... |
3,485,208 | <p>the reverse of the number <span class="math-container">$129$</span> is <span class="math-container">$921$</span> and these add to <span class="math-container">$1050$</span> which is divisible by <span class="math-container">$30$</span>. How many three-digit numbers have the property that, when added to their reverse... | The Demonix _ Hermit | 704,739 | <p>Let the number be <span class="math-container">$100a+10b+c$</span>.</p>
<p>Since the sum is divisible by <span class="math-container">$30$</span> , It should also be divisible by <span class="math-container">$10$</span> and <span class="math-container">$3$</span>.</p>
<p>Taking <span class="math-container">$a=1$</... |
1,560,549 | <p>Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. </p>
<p>I was thinking of doing the integral test to approximate this but I am unsure if this would work.</p>
| Claude Leibovici | 82,404 | <p><em>This answer if more than likely off-topic. In any manner, it is too long for a comment.</em></p>
<p>Ross Millikan gave a simple way to approximate the summation. Making the problem more general $$S=\sum_{i=2}^{a-1}\frac{a-i}{a+i}\approx \int_2^{a}\frac{a-i}{a+i}\,di=2+a (2 \log (a)+\log (4)-1)-2 a \log (a+2)$$ ... |
2,138,631 | <p>If you reverse the binary digits of a multiple of $3$, the result is always a multiple of $3$. The same is not true for $7$, but it does appear to happen more often than $\frac{1}{7}$ of the time. For fun, I'll call a number $n$ such that the reverse of the binary digits of $7n$ represents a multiple of $7$ a "seven... | Bram28 | 256,001 | <p>I have a suspicion that eventually this proportion will go down to where you intuit it should be, which is $\frac{1}{7}$, but that it is initially a good bit higher that, and that it wil take quite a while to converge to this value.</p>
<p>I have no proof for this, but I do have my reasons:</p>
<p>First let's note... |
426,579 | <p>I was doing some work with diagonalization of a matrix $A$ in order to find a matrix $P$ such that $\,P^{-1}AP\,$ was diagonal. In order to that I set $\;\lambda I_{n}=0\;$ and found the characteristic polynomial and its roots.</p>
<p>When I factored my characteristic polynomial I obtained $\;\lambda^2(\lambda-2),\... | Ben Passer | 79,968 | <p>A matrix is called diagonal if its off-diagonal entries ($a_{ij}$ for $i \not= j$) are all zero. This does not require the diagonal entries ($a_{jj}$) to be nonzero. For instance, the zero matrix is diagonal.</p>
|
4,191,936 | <p>I'm not a mathematician so please forgive me if I have a poor understanding of some math concept.</p>
<p>I see a lot of tutorial on how to generate fractals and Mandelbrot images from a pixel-wise perspective but I would like to know if it is possible to find some kind of parametric equation to see contours of image... | davidlowryduda | 9,754 | <p>The way those colors are given to the Mandelbrot set is to count the number of iterations of the Mandelbrot map it takes to get to a certain magnitude (which seems to be <span class="math-container">$2$</span> in this case). That is, if we define the map <span class="math-container">$f_c(z) = z^2 + c$</span>, then f... |
1,324,045 | <p>Here $(x_i) \in \ell^2$. All I have on paper right now is
$$\rVert Tx\rVert = \sqrt{\sum_{i=1}^{\infty} \frac{x_i ^2}{i^2}}$$</p>
<p>and I'm not sure about what's next. Please help. </p>
| WW1 | 88,679 | <p>The simplest bit of arithmetic that can be proved is that $2+2=4$</p>
<p>the number $1$ is assumed to exist, then the following are definitions of the symbols $2, 3 \text{ and } 4 $</p>
<p>$$ 1+1 \equiv 2$$
$$ 1+2=2+1 \equiv 3$$
$$ 1+3=3+1 \equiv 4$$</p>
<p>now we can prove</p>
<p>$$ 2+2 = 2 + (1 + 1) = (2+1) + ... |
1,324,045 | <p>Here $(x_i) \in \ell^2$. All I have on paper right now is
$$\rVert Tx\rVert = \sqrt{\sum_{i=1}^{\infty} \frac{x_i ^2}{i^2}}$$</p>
<p>and I'm not sure about what's next. Please help. </p>
| Red Banana | 25,805 | <p>It really depends on what you are calling a <em>proof</em>. The idea is quite universal, but the means to <em>prove</em> things may differ. For example, <a href="https://en.wikipedia.org/wiki/Principia_Mathematica" rel="nofollow noreferrer">it took $362$ pages in Russell's Principia Mathematica to prove that $1+1=2$... |
56,688 | <p>Please help me to prove tha
$$\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$$ converges and its limit is $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}$.</p>
| Sasha | 11,069 | <p>The following <a href="https://math.stackexchange.com/questions/55358/asymptotic-behavior-of-left-sum-limits-j-1n-frac1j-right-logn/55372#55372">question</a> addresses the convergence, so assume $\gamma_n$ converges and denote the limit as $\gamma$.</p>
<p>Now consider $\sum_{n=1}^{2m} (-1)^{n+1} \frac{1}{n}$ and ... |
56,688 | <p>Please help me to prove tha
$$\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$$ converges and its limit is $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}$.</p>
| robjohn | 13,854 | <p>The Mean Value Theorem says
$$
\frac{1}{k+1}<\log(k+1)-\log(k)<\frac{1}{k}
$$
Summing from $m$ to $n-1$ ($m < n$), we get
$$
\begin{align}
\sum_{k=m}^{n-1}\frac{1}{k+1}<\log(n)-\log(m)<\sum_{k=m}^{n-1}\frac{1}{k}
\end{align}
$$
Subtracting $\displaystyle{\sum_{k=m}^{n-1}\frac{1}{k+1}=\sum_{k=m+1... |
610,423 | <p>How can one construct a universal prefix-free Turing machine (TM)? By a universal prefix-free TM, I mean a prefix-free TM $U$ (that is, a TM whose domain is prefix-free) such that for every prefix-free partial recursive function $f$, there feasibly exists $\rho_f$ such that $U(\rho_fx) = f(x)$ for any $x$.</p>
<p>... | PyRulez | 49,592 | <p>Here is another way.</p>
<p>Let <span class="math-container">$p_f$</span> be a computable prefix code for partial recursive functions. For example, you could let <span class="math-container">$p_f$</span> be <span class="math-container">$0^{|d(f)|}1d(f)$</span>, where <span class="math-container">$d(f)$</span> is a ... |
3,226,962 | <p>I know that if <span class="math-container">$f$</span> is a polynomial and <span class="math-container">$f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$</span>, then <span class="math-container">$a$</span> is a root of multiplicity <span class="math-container">$k$</span>. Does this work for a differentiable function that is not... | Community | -1 | <p>Locally, a smooth (differentiable) function behaves like a polynomial, as shown by its Taylor's development. The multiplicity is given by the number of leading zero coefficients.</p>
<p>On the plot, the given function and it's second order Taylor approximation:</p>
<p><a href="https://i.stack.imgur.com/f0O15.png" ... |
1,722,670 | <p>In a 52-card deck, Jack is worth $1$ point, Queen $2$ points, King $3$ points, Ace $4$, and everything else is worth $0$.</p>
<p>How to calculate the probability you have a hand with sum $C\geq13$ points? where one hand is $13$ cards.</p>
| Henricus V. | 239,207 | <p>Using Cauchy condensation test, its convergence is equivalent to
$$ \sum_{n=1}^\infty 2^n \frac{2^n - 1}{2^{2n}} = \sum_{n=1}^\infty \frac{2^n - 1}{2^n}
$$
so...</p>
|
763,956 | <p>While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex algebras? Are there higher dimensional cyclotomic fields, and if so do they ... | rschwieb | 29,335 | <blockquote>
<p><em>Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex algebras?</em></p>
</blockquote>
<p>They can still be defined as elements satisfying $x^n=1$ for a nonnegative integer $n$. (I'm not sure what other definition you would use... )</p>
<p>Of course, that does... |
948,306 | <p>Assuming a standard English 26-letter alphabet, and $k<26$, what's the number of $k$-letter words where the letters don't repeat? So for example 'house' has no repeat letters, while 'rolls' has a repeat. </p>
<p>The way I solved this was to first select the $k$ letters which will make up the word, and then count... | user103828 | 103,828 | <p>Let $P_N$ be the probability of at least $4$ consecutive heads in $N$ tosses and $p$ be the probability of a head. Then conditioning on the first toss,
\begin{align*}
P_4 &=p^4 \\
P_5 &=(1-p)P_4+p^4= (2-p)p^4\\
P_6 &=(1-p)P_5+p(1-p)P_4+p^4=(3-2p)p^4 \\
P_7 &=(1-p)P_6+p(1-p)P_5+p^2(1-p)P_4+p^4=(4-3p)... |
3,299,532 | <p>Could you please give me a hint on how to prove the inequality below for <span class="math-container">$k \geq 3$</span>?</p>
<blockquote>
<p><span class="math-container">$$\frac{\sqrt{k(k+1)} }{k-1} \leq1 + \frac{2}{k} + \frac{3}{4k^2} $$</span> </p>
</blockquote>
<p>Thank you in advance.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>We have
<span class="math-container">$$\left(1+\frac{2}{k}+\frac{3}{4k^2}\right)^2-\frac{k(k+1)}{(k-1)^2}=1/16\,{\frac {16\,{k}^{5}-24\,{k}^{4}-64\,{k}^{3}+{k}^{2}+30\,k+9}{{k}
^{4} \left( k-1 \right) ^{2}}}
\geq 0$$</span> for <span class="math-container">$k\geq 3$</span></p>
|
2,467,608 | <p>For a vector-valued function $f:c\in E\subseteq \Bbb R^n\to\Bbb R^m$, I want to prove that the linear transformation $T$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T(h)}{|h|}=0$ is unique.</p>
<p>My attempt:
Suppose $T_1,T_2:\Bbb R^n\to\Bbb R^m$ such that $\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)-T_... | Elias Costa | 19,266 | <p>Let$T_1(v)\neq T_2(v)$ for a fixed $v\in \mathbb{R}^n$. Then
$$
T_1\left(\frac{1}{\|v\|}v\right)- T_2\left(\frac{1}{\|v\|}v\right)\neq 0
$$ We know that
$$
\lim_{t\to 0}\frac{f(c+tv)-f(c)-T_1(tv) }{\left\|tv \right\|}=0
\\
\lim_{t\to 0}\frac{f(c+tv)-f(c)-T_2(tv) }{\left\|tv \right\|}=0
$$
implies
$$
\lim_{t\to 0}... |
868,643 | <p>How do you integrate the following by using Euler's formula, without using integration by parts? <span class="math-container">$$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}d\theta$$</span></p>
<p>I did integrate it by parts, by writing the <span class="math-container">$3$</span> in the numerat... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>When you have an expression with a squared denominator, you could think that the solution is of the form $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}~d\theta=\frac{a+b\sin \theta+c\cos \theta}{3\cos {\theta}+4}$$ Differentiate the rhs and identify terms. You will... |
2,372,149 | <p>What is the sum of the absolute values of all the $x$-s that satisfy the equation:$$(2 + \sqrt 3)^x + (2 - \sqrt 3)^x=4$$</p>
| Michael Hoppe | 93,935 | <p>Let $f(x)=(2 + \sqrt 3)^x + (2 - \sqrt 3)^x-4$. Now show that $f$ decreases for negative $x$ and increases for positive $x$.</p>
<p>Obviously $1$ is a zero of $f$. Then from the symmetry of $f$, namely $f(-x)=f(x)$, it follows that $-1$ is the only other zero of $f$, so their absolute values sum up to $2$.</p>
|
3,962,608 | <p>A travel agency offers the chance to its customers to win vouchers for 2 free excursions during the Covid crisis. Out of the x total customers, y will be randomly chosen to participate in the first excursion and z in the second. In how many of all possible draws we will have a total of w customers who will participa... | Phicar | 78,870 | <p>You can choose the lucky <span class="math-container">$w$</span> first in <span class="math-container">$\binom{x}{w}$</span> ways, then from the remainder <span class="math-container">$x-w$</span> you choose the ones that were in the first excursion in <span class="math-container">$\binom{x-w}{y-w}$</span> and now f... |
98,527 | <p>Let $I\subset\mathbb{R}$ be a compact interval and let $J$ denote its interior.<br>
Consider $f:J\to\mathbb{R}$ being continuous.</p>
<ol>
<li>Under which conditions does the following statement hold?<br>
$$
\text{There exists a continuous extension $g:I\to\mathbb{R}$ of $f$.}\tag{A}
$$</li>
<li>Is boundedness of $... | Alex Youcis | 16,497 | <p>Call a continuous function $f:X\to \mathbb{R}$ (where $X$ is some metric spaces) <em>Cauchy continuous</em> if $f$ takes Cauchy sequences in $X$ to Cauchy sequences in $\mathbb{R}$. It is then a classic theorem that a continuous function $f:X\to \mathbb{R}$ admits an extension $\widetilde{f}:\overline{X}\to \mathbb{... |
754,277 | <p>I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube. </p>
<p>For example:</p>
<p>2^2 -- a 2 by 2 square</p>
<p>2^3 -- a 2 by 2 by 2 cube</p>
<p>I've been puzzling over how something would look when you make it to the power of a fraction.... | Andrés E. Caicedo | 462 | <p>Cohen's result is that from a certain set of axioms ($\mathsf{ZFC}$) we cannot prove the continuum hypothesis. Gödel's result is that from the same set of axioms, we cannot refute the continuum hypothesis. This only means that the set of axioms under consideration is not strong enough to settle this question. If you... |
754,277 | <p>I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube. </p>
<p>For example:</p>
<p>2^2 -- a 2 by 2 square</p>
<p>2^3 -- a 2 by 2 by 2 cube</p>
<p>I've been puzzling over how something would look when you make it to the power of a fraction.... | user143522 | 143,522 | <p>You have a mistake here: "if we cannot find such a set, then we can only conclude that such a set does not exist".</p>
<p>That's not right. The fact that you can't find a set does not mean that it does not exist. That's like the planets located beyond the observable universe: you can't prove if there are some peopl... |
301,259 | <p>Problem : How many ways are there to make $n$ by adding $k$ non-negative integers, where order matters. Suppose $n=4$ and $k=3$. There are 15 solutions using $0, 1, 2, 3, 4$:</p>
<p>$(0,0,4), (0,1,3), (0,2,2), (0, 3, 1), (0, 4, 0), (1, 0, 3), (1, 1, 2), (1, 2, 1), (1, 3, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0), (3, 0, ... | Community | -1 | <p>The classical solution goes like this: $n=4$ is made up of four objects, called "stars":</p>
<pre><code>* * * *
</code></pre>
<p>Our goal is to group these stars into three summands. We can do so by placing "bars" to separate the stars. For example, the solution "1 0 3" is</p>
<pre><code>*||* * *
</code></pr... |
195,464 | <p>Random variable <strong>X</strong> (as a measureable function from some probability space to the reals) has support <em>S</em> and image <em>T</em>. True or False: </p>
<p>i.) <em>S</em> is always a subset of <em>T</em>.</p>
<p>ii.) <em>(S,E,dF)</em> is the induced probability space with: sample space <em>S</em>, ... | user782220 | 21,556 | <p>Let $L=(\gamma(1+\alpha)/2)^{1/(1-\alpha)}$.</p>
<p>Suppose that $|x_k| < L = (\gamma(1+\alpha)/2)^{1/(1-\alpha)}=((\gamma(1+\alpha)/2)^{\alpha/(1-\alpha)})^{1/\alpha}=((\gamma(1+\alpha)/2)^{-1}(\gamma(1+\alpha)/2)^{1/(1-\alpha)})^{1/\alpha}=((\gamma(1+\alpha)/2)^{-1}L)^{1/\alpha}$</p>
<p>from which we conclude... |
2,351,690 | <p><strong>Background Information:</strong></p>
<p>I am using the book Partial Differential Equations by Walter A. Strauss. In chapter $5$ section $6$ the author solves the diffusion equation with sources at the endpoints.
$$u_t = k u_{xx} \ \ 0<x<l, t>0$$
$$u(0,t) = h(t) \ \ \ \ \ u(l,t) = j(t)$$
$$u(x,0) =... | JJacquelin | 108,514 | <p>With the method of separation of variables and Fourier series for the fitting to the boundary equations :</p>
<p><a href="https://i.stack.imgur.com/tU3gc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tU3gc.png" alt="enter image description here"></a></p>
<p>Or, on another equivalent form :
$$u... |
1,528,979 | <p>As tangent is a line passing through a curve in one point, and secant is a line passing through a curve in two points, is sine also something like this?</p>
| Henricus V. | 239,207 | <p>Tangent has two meanings:</p>
<ol>
<li><p>The $\tan: \mathbb{R} \to \mathbb{R}$ function.</p></li>
<li><p>A straight line that intersects a curve at one and only one point.</p></li>
</ol>
<p>The $\sin$ function represents the $y$-coordinate of a point on a circle.</p>
|
1,901,722 | <p>I just stared a new math course, and we began by reviewing metric spaces. The professor gave us the set $X = (-\infty,\infty) $ , so all real numbers, and asked us to come up with any metric for it. When we suggested $|x-y|$, the simplest we could think of, he said that we were wrong because this failed to work fo... | YTS | 126,222 | <p>You must think that $\infty$ and $-\infty$ are indeed points and they are not just concepts. Your professor is asking about a metric on $X$ that preserve the usual topology on $\mathbb{R}$ and such that preserve our usual understanding about infinities, I mean that $-\infty<x<\infty$. The trick is to find some... |
112,048 | <p>This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giant-component" and "fully connected"?
e.g. a detailed picture for say, $p_n=\frac{(\log n)^\gamma}{n}$, with $0<\gamma<1$</p>
| Joseph O'Rourke | 6,094 | <p>This may not satisfy your needs, but this recent (Fall 2012) paper, or its references might help:</p>
<blockquote>
<p>"Lack of Hyperbolicity in Asymptotic Erdös-Renyi Sparse Random Graphs,"
by Onuttom Narayan, Iraj Saniee, Gabriel H. Tucci. (<a href="http://arxiv.org/abs/1009.5700" rel="nofollow noreferrer... |
1,446,439 | <p>Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number</p>
<p>Note: In this problem a labeling of rational numbers by positive integers is used. Such labellings do exist because $\mathbb{Q}$ is countable and we choose any of them. The answer does not depend on the choice.</p>
<p>Def: $c$ is ... | CIJ | 159,421 | <p>Let $c$ be an arbitrary real number, let $N$ be an arbitrary natural number and let $\varepsilon>0$ be arbitrary. If none of the rationals $x_{N+1},x_{N+2},x_{N+3},\ldots$ belongs to the open interval $(c-\varepsilon,c+\epsilon)$ then $(c-\varepsilon,c+\varepsilon)\cap\mathbb Q$ is finite, which cannot happen sin... |
1,446,439 | <p>Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number</p>
<p>Note: In this problem a labeling of rational numbers by positive integers is used. Such labellings do exist because $\mathbb{Q}$ is countable and we choose any of them. The answer does not depend on the choice.</p>
<p>Def: $c$ is ... | TravisJ | 212,738 | <p>Let $c$ be any real number. What you want to show is that there is some subsequence that converges to $c$. Let $\epsilon_{n}=\frac{1}{n}$ (really, anything that is positive and converges to $0$). Now, pick $n_1$ so that $x_{n_1}\in (c-\epsilon_1, c+\epsilon_1)$. Then, pick $n_2$ so that $x_{n_2}\in (c-\epsilon_2... |
3,272,736 | <p>For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2)</p>
<p>What is Cov(X, Y) equal to?</p>
<p>What I tried was:
E[XY] - E[X]E[Y]</p>
<p>Where E[X] = 5.45 and E[Y] = 1.4</p>
<p>But for E[XY] I'm a bit clueless. I've considered:
<span class="math-container">$$ \int_1^{9.9}\int_{-\infty}^{\infty} xx \frac{1}{9.9-1} ... | callculus42 | 144,421 | <p>Your calculation is right. <span class="math-container">$\mathbb E(X\cdot Y)=\mathbb E_X(X\cdot \mathbb E_{Y|X}[Y])=7.63$</span></p>
<p>You´re right as well that <span class="math-container">$\mathbb E(X)\cdot \mathbb E(Y)=7.63$</span>. That means that <span class="math-container">$Cov(X,Y)=0$</span>.</p>
<p>This ... |
1,267,956 | <p>I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. </p>
<p>I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence why the integral is improper for these boundaries).</p>
<p>I am now trying to evaluate the limits to see if it ... | Yes | 155,328 | <p>Let $b > 16.$ We claim that $\int_{b}^{100}\frac{1}{x^{1/4} - 2} dx$ does not converge as $b \to 16+.$</p>
<p>But, since
$$\int \frac{1}{x^{1/4} - 2} dx \simeq \frac{4x^{3/4}}{3} + 4x^{1/2} + 16x^{1/4} + 32 \log |2 - x^{1/4}|,$$
we have
$$\frac{-4b^{3/4}}{3} - 4b^{1/2} - 16b^{1/4} + 32 \log |2 - b^{1/4}| \to \f... |
2,594,453 | <p>I'm at the very beginning in algebra and also I don't speak English much. Maybe this question is very simple but I can't understand how to determine parameters in polynomials in order to find out real, rational or integer roots. Here's the polynomial:</p>
<p>$$x^{3} - 33x^{2} + 315x -k$$</p>
| anomaly | 156,999 | <p>Let $P(x)$ denote the given polynomial. Since the coefficients of $P$ are real, the non-real roots of $P$ occur in conjugate pairs. Since $P$ has $3$ roots, that means that $P$ has either $1$ or $3$ real roots.</p>
<p>But that's counting roots with multiplicities; for example, $x(x-1)^2$ has a root of order $1$ at ... |
3,480,368 | <p>Let <span class="math-container">$f$</span> be a weakly increasing step function. <span class="math-container">$g(x)=ax$</span> be a weakly increasing linear function.</p>
<p>Let <span class="math-container">$h=f+g$</span>. What is <span class="math-container">$h^{-1}$</span>?</p>
<p><span class="math-container">$... | CyclotomicField | 464,974 | <p>An inverse function cannot be constant on some interval. This is easy to see graphically in the single variable case as the inverse function is merely the reflection about the line <span class="math-container">$y=x$</span>. If the graph of an inverse function is constant on a continuous region then when we reflect a... |
983,930 | <p>Let us consider an equation $x^3+10x^2-100x+1729=0$. Will this equation have at least one complex root having modulus $>12$?</p>
| Antony | 180,827 | <p>$7^{20}=(7^5)^4=((16807)^2)^2\equiv((807)^2)^2\equiv(651249)^2\equiv249^2\equiv62001\equiv1\pmod {1000}
$
<br> Then we write $7^{341}=7*(7^{20})^{17}\equiv7*1^{17}\equiv7\pmod {1000}$</p>
|
983,930 | <p>Let us consider an equation $x^3+10x^2-100x+1729=0$. Will this equation have at least one complex root having modulus $>12$?</p>
| Sarastro | 31,047 | <p>To find $7^{341} (\bmod 1000)$, it suffices to work out $7^{341} (\bmod 125)$ and $7^{341} (\bmod 8)=-1$. The former can be done with successive squaring, as you mentioned.</p>
|
983,930 | <p>Let us consider an equation $x^3+10x^2-100x+1729=0$. Will this equation have at least one complex root having modulus $>12$?</p>
| lab bhattacharjee | 33,337 | <p>$$7^2=50-1$$</p>
<p>$$7^{20n+1}=7(7^2)^{10n}$$</p>
<p>Now $(7^2)^{10n}=(50-1)^{10n}=(1-50)^{10n}$</p>
<p>Using Binomial Theorem,
$(7^2)^{10n}=1-\binom{10n}150+\binom{10n}2(50)^2\cdots+(50)^{2n}$</p>
<p>$\implies(7^2)^{10n}\equiv1-500n+\dfrac{10n(10n-1)}250^2\pmod{1000}\ \ \ \ (1)$ for $n\ge2$</p>
<p>Again, $\df... |
1,794,039 | <p>This is the Guess-The-Number game with a twist!</p>
<h2>Variant 1</h2>
<blockquote>
<p>Take any positive integer <span class="math-container">$n$</span>.</p>
<p>The game-master chooses an <span class="math-container">$n$</span>-bit integer <span class="math-container">$x$</span>.</p>
<p>The player makes queries one ... | Anon | 245,264 | <p>$$Z(G) = \{g \in G \mid \forall a \in G: ga = ag\} = \bigcap_{a \in G} \{g \in G \mid ga = ag\} = \bigcap_{a \in G} C(a)$$</p>
|
3,979,598 | <p>Quick question. I was wondering whether the following limit is simply equal to zero or it is rather indeterminate:</p>
<p><span class="math-container">$$\lim_{x\to0}\frac0x$$</span></p>
<p>For more detail: I was studying the differentiability of the function <span class="math-container">$\lfloor x\rfloor\sin(\pi x)^... | PrincessEev | 597,568 | <p>The notion of an "indeterminant form" follows moreso from taking the limit of a ratio as the ratio of limits:</p>
<p><span class="math-container">$$\lim_{x \to a} \frac{f(x)}{g(x)} \stackrel{\text{becomes}}{\longrightarrow} \frac{\lim \limits_{x \to a} f(x)}{\lim \limits_{x \to a} g(x)}$$</span></p>
<p>...... |
368,293 | <p><a href="http://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx" rel="noreferrer">P-Series Reference</a></p>
<p><a href="http://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx" rel="noreferrer">Alternating Series Test Reference</a></p>
<p>$$ \sum_{i=0}^\infty \frac{(-1)^n}{n} $$</p>
<p>This ... | amWhy | 9,003 | <p>What you are noting is that the series $$\sum_{i=0}^\infty \,\frac{(-1)^n}{n}\quad {\bf {converges},}$$ as you found by the <em>alternating series test</em>, <em>but does not converge <strong>absolutely</em></strong>: $$\sum_{i=0}^\infty \,\left|\frac{(-1)^n}{n}\right| \quad = \quad \sum_{i = 0}^\infty\,\frac 1n\qua... |
1,275,779 | <p>I am taking Abstract Algebra now and it seems most of its examples come from elsewhere. For example, I know what permutations are in counting, but the kind Algebra studies seem to be somewhat different. And to me these concepts Algebra comes bundled with is the most difficult part of studying it. It's like studying ... | Community | -1 | <p>There might be, but a huge chunk of abstract algebra IS those examples. Learning about permutations (which are exactly what you know) with this language IS abstract algebra. A huge part of abstract algebra is knowing many different examples, things that you might not see elsewhere.</p>
<p>Although for a gentle intr... |
1,514,981 | <p>I have one question as following:
Can we move f(x) out of $\int_{-\infty}^{\infty}f(x)sinc(x)dx$?</p>
<p>If $f(x')\int_{-\infty}^{\infty}sinc(x)dx$=$\int_{-\infty}^{\infty}f(x)sinc(x)dx$?</p>
| El Spiffy | 287,369 | <p>Arctan is only true from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ (I.e the domain of arctan)</p>
|
1,787,867 | <p>Prove that $\dfrac{2\cos x}{\cos 2x + 1 }= \sec x$. </p>
<p>So far I have:
$\dfrac{2\cos x}{\cos 2x + 1 }= \dfrac 1 {\cos x}$</p>
<p>Where do I go from here?</p>
| sina | 323,942 | <p>We know that</p>
<p>$$\cos2x=2\cos^2x-1$$ </p>
<p>When the above is plugged into the denominator, we have</p>
<p>$$\frac{2\cos x}{\cos 2x + 1 }= \frac {2\cos x}{2\cos^2x }=\frac1{\cos x} = \sec x$$</p>
|
1,436,715 | <p>After reading some text books about Brownian Motion i often encountered the following object
$$
\sup_{t \in [0, T]} B_t,
$$
where $(B_t)_{t \geq 0}$ is a Brownian Motion. <br/>
But how do i see that this object is measurable? The problem is that the supremum is taken over an uncountable set. Do i miss something triv... | Mankind | 207,432 | <p>Typically a Brownian motion is defined to have continuous sample paths. If you take some countable, dense subset $S\subset [0,T]$, you then have</p>
<p>$$\sup_{t\in S} B_t = \sup_{t\in [0,T]} B_t$$</p>
<p>by continuity of $B_t$.</p>
|
996,868 | <p>I am currently learning about vector spaces and have a slight confusion.</p>
<p>So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list of axioms.</p>
<p>I know that a subspace is created from the subset of a vector space and also defined by 3 pr... | NicNic8 | 24,205 | <p>Yes, any vector space is a subspace of itself.</p>
|
3,645 | <p>Being motivated by <a href="https://math.stackexchange.com/questions/3618/solving-an-equation-with-irrational-exponents">this post</a>, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving <a href="http://en.wikipedia.org/wiki/Transcendental... | Bill Dubuque | 242 | <p>Intuition may be misleading here. In fact transcendental cases are often much easier than the integer Diophantine case. For example below is a table listing the known decidability results in various rings for Hilbert's tenth problem and the full first order theory, excerpted from Bjorn Poonen's interesting paper <a ... |
3,645 | <p>Being motivated by <a href="https://math.stackexchange.com/questions/3618/solving-an-equation-with-irrational-exponents">this post</a>, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving <a href="http://en.wikipedia.org/wiki/Transcendental... | IV_ | 292,527 | <p>A general method for solving a given equation $H(x)=0$ is to apply the compositional inverse $H^{-1}$ of $H$: $x=H^{-1}(0)$. In general, $H$ and $H^{-1}$ are correspondences. But often it is possible to split the problem into subproblems where $H$ and $H^{-1}$ are functions.</p>
<p>For applying this method, $H$ and... |
521,739 | <p>Consider the r-subsets of {1,2,...,n} in lexicographic order. What are the first (n-r+1) r-subsets?</p>
<p>The answer is 12...r, 12...(r-1)(r+1),...,12...(r-1)n </p>
<p>I don't understand where the 12 is coming from.
In my book it says that 12...r is the first r-subset in the lexicographic ordering and (n-r+1)(n-... | Brian M. Scott | 12,042 | <p>For a concrete example take $n=9$ and $r=4$, so that we’re looking for the first six $4$-subsets of $[9]=\{1,\ldots,9\}$ in lexicographic order. If $A$ and $B$ are $4$-subsets of $[9]$, we determine which comes before the other as follows. Let $A=\{a_1,a_2,a_3,a_4\}$ and $B=\{b_1,b_2,b_3,b_4\}$, where $a_1<a_2<... |
1,302,879 | <p>There is a question in our analysis book and I have lots of problem with it. It says that:</p>
<p>set</p>
<p>$$
f(x)= \begin{cases}
0 \quad & x\leq0\\
\sin(\frac{\pi}{x}) \quad & x>0
\end{cases}
$$</p>
<p>$$
g(x)=
\begin{cases}
0 \quad & x\leq0\\
1 \quad & x>0.
\end{cases}
$$</p>
<p>... | Tim Raczkowski | 192,581 | <p>Consider
$$G(x)=\int_{-\infty}^xg(t)\,dt.$$</p>
<p>For $x\ne0$, $G'(x)=g(x)$, but $G'(0)$ is not defined.</p>
|
3,860,441 | <p>I'm taking up a probability course and my teacher, when explaining the fondamental properties of probability, listed this one:
<span class="math-container">$$\text{If } A \text{ and }B \text{ are mutually exclusive events, then }P(A \cup B) = P(A) + P(B)$$</span></p>
<p>Then, later, under an "other properties&... | Cornel Stoica | 835,079 | <p>This is best viewed by drawing some venn diagrams.
In the first case, if it is said that they are mutually exclusive (never touching or intersecting) then we can simply add up the elements in A and B. The venn diagram for this would look like two circles labeled A and B separated from each other.</p>
<p>Otherwise, i... |
463,500 | <p><a href="http://en.wikipedia.org/wiki/Theory_%28mathematical_logic%29#Theories_associated_with_a_structure" rel="nofollow">According</a> to Wikipedia, every $\sigma$-structure $A$ is associated with a collection of atomic and negated atomic sentences called its <em>diagram</em>. Here's the precise definition.</p>
<... | Alex Kocurek | 74,375 | <p>As William has correctly noted, the diagram certainly <em>does not</em> determine the structure in question. True, it tells you every atomic thing that's true or false about the element of the original model (say $\cal{A}$), but there are other models with <em>more</em> structure which could satisfy that diagram (fo... |
956,186 | <p>I don't understand what I should calculate if is said: "find the unique x", e.g. in the following exercise:</p>
<p>"Find the unique x in the interval [0,pi] with cos(x) = 0,5 * sqrt(3)"</p>
| Paul | 17,980 | <p>"find the unique $x$" means </p>
<p><strong>1,</strong> Such $x$ must exist; </p>
<p><strong>2,</strong> it is the only one.</p>
|
94,481 | <p>I have some plots in an old Mathematica notebook that I'd like to re-create in Matplotlib (for stylistic consistency with the rest of the plots in my thesis).</p>
<p>Unfortunately, I've lost the data that I used to generate these plots. Since the plots are still intact, is it possible to extract the coordinates of... | march | 29,734 | <p>As explained by <code>anderstood</code>, if you don't have the plots named, you need to copy and paste the figure into an input cell and type, say, <code>fig =</code> in front of the figure, then evaluate.</p>
<p>Suppose you generated a <code>MoneyPlot</code> named <code>fig</code>. Then, you can extract the data m... |
1,630,335 | <p>Show that the list $(1,2), (3,5)$ is a basis for $\mathbb R^2$.</p>
<p>In order to show it is a basis, I have show that the list is linearly independent and that the list spans $\mathbb R^2$. </p>
<p>So I understand how to show linear independence (you simply set up a system of equations). However, I am uncertain ... | Mathmo123 | 154,802 | <p>To show that $(1,2),\ (3,5)$ span $\mathbb R^2$, you need to show that given $(a,b)\in \mathbb R^2$, we can find $\alpha,\beta\in\mathbb R$ such that $$(a,b) =\alpha(1,2)+\beta(3,5)$$
or equivalently, we need to solve the simultaneous equations
$$a = \alpha + 3\beta\\ b=2\alpha + 5\beta.$$
One can manually show that... |
1,630,335 | <p>Show that the list $(1,2), (3,5)$ is a basis for $\mathbb R^2$.</p>
<p>In order to show it is a basis, I have show that the list is linearly independent and that the list spans $\mathbb R^2$. </p>
<p>So I understand how to show linear independence (you simply set up a system of equations). However, I am uncertain ... | Dietrich Burde | 83,966 | <p>The determinant of $$\begin{pmatrix}1&3\\2&5\end{pmatrix}$$
is nonzero over any field $F$. Hence the vectors are a basis for $F^2$. In particular this is true for $F=\mathbb{R}$.</p>
|
362,252 | <p>Consider the advection equation $u_t + c\, u_x =0$ with initial condition
$u(x,0)=f(x)$, so that $u(x,t) = f(x-c\,t)$ is the solution.</p>
<p>It is asked to show that
\begin{equation}
\int_{-\infty}^{\infty} u^2(x,t)\, dx = \int_{-\infty}^{\infty} u^2(x,0)\, dx =\int_{-\infty}^{\infty} f^2(x)\, dx
\end{equation}
fo... | simon | 47,709 | <p>$$
\frac{\partial}{\partial t} \int_{-\infty}^{\infty}u^2dx=
\int_{-\infty}^{\infty} 2\,u \, u_t\,dx
$$
using $u_t=-c\,u_x$ gives
$$
\int_{-\infty}^{\infty} 2\,u \, u_t\,dx = -2\,c\,\int_{-\infty}^{\infty} u \, u_x\,dx
$$
by integration by parts this becomes
$$
-2\,c\,\int_{-\infty}^{\infty} u \, u_x\,dx
=-2c\left[ ... |
3,846,739 | <p>An urn contains one black ball and one white ball. At every time step, a ball is chosen randomly from the urn and returned with another new ball of the same color. (So at time <span class="math-container">$ t $</span> there are a total of <span class="math-container">$ t+2 $</span> balls). What is the probability th... | Rezha Adrian Tanuharja | 751,970 | <p>At first there are <span class="math-container">$2$</span> balls to pick, then there are <span class="math-container">$3$</span>, then <span class="math-container">$4$</span>, so on until there are <span class="math-container">$t+1$</span>. There are <span class="math-container">$(t+1)!$</span> possibilities.</p>
<p... |
244,046 | <p>Let $X$ bet a metric space with metric $d$.</p>
<p>Prove that $$p(x,y) = \frac{d(x,y)}{1 + d(x,y)}$$ defines a metric on $X$.</p>
<ol>
<li><p>Show $p(x,y)$ is is bounded by $1$. (i.e. $p(x,y) \le 1$ for all $x,y$).</p></li>
<li><p>Given $x \in X$, determine $B_1^p$.</p></li>
</ol>
<p>I can prove it is a metric, I... | Michael Hardy | 11,667 | <p>The fact that $\dfrac{n}{1+n}\to1$ as $n\to\infty$ does not prove that $\dfrac{n}{1+n}\le 1$. And you don't need limits for that. If $A,B$ are positive numbers and $A<B$, then $\dfrac AB\le 1$.</p>
<p>The usual definitions of metrics do not allow points to be infinitely far apart, so you don't really need to s... |
4,226,115 | <p>I was going through some practice problems for an upcoming calculus exam and I came across the question: "Prove the following statement: if <span class="math-container">$\lim _{n\to \infty }a_n = L $</span> and <span class="math-container">$\lim _{n\to \infty }a_n = M$</span>, then <span class="math-container">... | user | 505,767 | <p>As noticed your proof is not correct. Assuming by contradiction as you did <span class="math-container">$L\neq M$</span>, we can proceed by the definition</p>
<p><span class="math-container">$$\lim _{n\to \infty }a_n = L \iff \forall\varepsilon \quad \exists n_L \quad \forall n>n_L \quad |a_n-L|<\varepsilon$$<... |
194,803 | <p>I ran into some problems while doing an exercise. The problem goes as follows:</p>
<p>Suppose we have two random independent variables $X$ and $Y$. Both are distributed normally with parameters $(0, \sigma^2)$. $\mathbb{P}(dx)=\frac{1}{\sigma\sqrt{2\pi}} \exp^{-\frac{x}{2\sigma^2}}dx$. For $\gamma \in \mathbb{R}$, ... | Michael R. Chernick | 30,995 | <p>To check independence of U and V you actually need $ E(UV)=E(U)E(V)$. Since U and V are both linear combinations of X and Y even knowing that X and Y are independent does not make the independence of U and V obvious.</p>
<p>Now E(UV)= E([X cosγ - Y sinγ][X sinγ + Ycosγ]) expanding this we get</p>
<p>E(X$^2$ sinγ... |
4,354,126 | <p>I know that, for a complex square matrix <span class="math-container">$A$</span>, there exists some non-null vector <span class="math-container">$v$</span> such that <span class="math-container">$Mv=0$</span> if and only if the determinant, <span class="math-container">$\det(M)=0$</span>. That is, <span class="math-... | nilsw | 729,497 | <p>The easiest way I know is to use the multiplicativity of the determinant and Cramers rule. More precisely: Let <span class="math-container">$I_n$</span> denote the <span class="math-container">$n\times n$</span> identity matrix.</p>
<p>For any two matrices <span class="math-container">$A,B\in M_{n,n}(K)$</span> it h... |
2,202,518 | <p>I'm trying to do exercise 9 of section 15.5 in the book Abstract Algebra written by Dummit and Foote, whose aim is to establish a homeomorphic map between the elements in the fiber over $(p)$ of the Zariski continuous map from $\mathrm{Spec}(\mathbb{Z} [x])$ to $\mathrm{Spec}(\mathbb{Z})$ and the elements in $\mathr... | AlexT | 428,860 | <p>You have a map $\phi: \mathbb{Z_7} \rightarrow M_2(\mathbb{Z_7})$ such that $\ f \mapsto f \begin{pmatrix} 0 & 2 \\ 2 & 2 \end{pmatrix}$. Basically you evaluate all possible linear combinations of the matrix $\begin{pmatrix} 0 & 2 \\ 2 & 2 \end{pmatrix}$, which we'll call $A$ for brevity, as f varie... |
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