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 |
|---|---|---|---|---|
33,702 | <p>The code that I have written has an unintended consequence that I'm not sure how to get around. I want 3 rotation transforms to be applied simultaneously to 1 graphics object. Instead, I get 3 separate separate copies of the graphics object, one per transformation.</p>
<p>The documentation does state that this wi... | Alexey Popkov | 280 | <p>The problem arises when <code>function</code> returns a number smaller than <code>$MinMachineNumber</code>:</p>
<pre><code>function[t_] := Exp[-9 t^2];
LogLogPlot[function[t], {t, 8.8718, 8.872}, PlotRange -> All,
GridLines -> {{{Sqrt[Log[1/$MinMachineNumber]]/3,
Directive[Thick, Dashed]}}, None}]
Sho... |
4,213,207 | <p>I am trying to find all homomorphisms from <span class="math-container">$\mathbb{Z}_{20}$</span> to <span class="math-container">$\mathbb{Z}_8$</span>. I understand how to do it - one completely determines any homomorphism, say <span class="math-container">$\phi$</span>, by computing multiples of <span class="math-c... | Dave | 334,366 | <p>You are correct about looking at <span class="math-container">$\phi(1)$</span>: since <span class="math-container">$1$</span> generates the domain (i.e. <span class="math-container">$\mathbb Z_{20}$</span>), you can just look at where <span class="math-container">$1$</span> gets sent under the homomorphism. The only... |
4,213,207 | <p>I am trying to find all homomorphisms from <span class="math-container">$\mathbb{Z}_{20}$</span> to <span class="math-container">$\mathbb{Z}_8$</span>. I understand how to do it - one completely determines any homomorphism, say <span class="math-container">$\phi$</span>, by computing multiples of <span class="math-c... | Geoffrey Trang | 684,071 | <p>We need to find how many <span class="math-container">$i \in \mathbb{Z}_{8}$</span> satisfy the relation <span class="math-container">$20i=0$</span>. There are only <span class="math-container">$8$</span> cases to consider:</p>
<ol>
<li><span class="math-container">$i=0$</span> works (<span class="math-container">$0... |
2,894,126 | <blockquote>
<p>$$\int \sin^{-1}\sqrt{ \frac{x}{a+x}} dx$$</p>
</blockquote>
<p>We can substitute it as $x=a\tan^2 (\theta)$ . Then:</p>
<p>$$2a\int \theta \tan (\theta)\sec^2 (\theta) d\theta$$</p>
<p>Using integration by parts will be enough here. But I wanted to know if this particular problem can be solved by... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Let $\sin^{-1}\sqrt{\dfrac x{x+a}}=y\ge0\implies\tan^2y=\dfrac xa\implies y=\tan^{-1}\sqrt{\dfrac xa}$</p>
<p>$$\int\sin^{-1}\sqrt{\dfrac x{x+a}}dx=\int\tan^{-1}\sqrt{\dfrac xa}dx$$</p>
<p>$$=\tan^{-1}\sqrt{\dfrac xa}\int dx-\int\left(\dfrac{d \tan^{-1}\sqrt{\dfrac xa}}{dx}\int dx\right)dx$$</p>
<p>... |
1,729,308 | <p>The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is </p>
<p>$a)$ $3$</p>
<p>$b)$ $4$</p>
<p>$c)$ $5$</p>
<p>$d)$ $6$</p>
<p>My approach : I used the formula for the first $n$ terms of an A.P. ... | Steven Alexis Gregory | 75,410 | <p>\begin{align}
n^2 +(a-1)n - 153 &= 0 \\
(n+u)(n-v) &= 0 \\
n^2 + (u-v)n - uv &= 0
\end{align}</p>
<p>So you need two positive integers, say $u>v>0$, such that $uv=153$ and $u-v=a-1$. Which will have solution $n=v$.</p>
<p>There aren't that many possibilities</p>
<p>\begin{array}{|rr|r|r... |
331,654 | <p>After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing below. </p>
<blockquote>
<p>Let $\mathscr{T}$ be the collection of subsets of $\Bbb R$ consisting
of $\emptyset, \Bb... | Brian M. Scott | 12,042 | <p>(b) Consider $\bigcup\{(r,\to):r\in\Bbb Q\text{ and }r>\sqrt2\}$.</p>
<p>(f) Consider the set $\mathscr{B}=\{(r,\to):r\in\Bbb Q\}$: is this a base for $\mathscr{T}$?</p>
<p>(g) A small correction: $\{\Bbb R\}$ is a finite cover of the space, as is any finite set of open sets that includes $\Bbb R$. However, $\{... |
132,266 | <p>I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. </p>
<blockquote>
<p>Does there exist software to compute symmetric powers of Schur polynomials? </p>
</blockquote>
<p>I am seeking such software in the hopes of computing the characters of represe... | Per Alexandersson | 1,056 | <p>If I am not mistaken, you are interested in the <a href="https://www2.math.upenn.edu/%7Epeal/polynomials/plethysm.htm" rel="nofollow noreferrer">plethysm</a>
<span class="math-container">$$
h_k[ s_\lambda]
$$</span></p>
<p>If you prefer Mathematica, I have a <a href="https://github.com/PerAlexandersson/Mathematica-... |
2,678,077 | <p>I need to solve this differential equation:
$$\frac{du}{dr}=\frac{4+\sqrt{r}}{2+\sqrt{u}}$$</p>
<p>I did it and got
$$u=\frac{2r^{\left(\frac{3}{2}\right)}}{3\sqrt{u}+2}+\frac{4r}{\sqrt{u}+2}+C$$ but my homework system is marking this as wrong. Why is that?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>you must write $$(2+\sqrt{u})du=(4+\sqrt{r})dr$$ and integrate this!
the solution should be $$4\,r+2/3\,{r}^{3/2}-2\,u \left( r \right) -2/3\, \left( u \left( r
\right) \right) ^{3/2}+{\it \_C1}=0
$$</p>
|
4,634,180 | <p><span class="math-container">$$\int \frac{\sin^2(x)dx}{\sin(x)+2\cos(x)}$$</span></p>
<p>I tried to use different substitutions such as <span class="math-container">$t=\cos(x)$</span>, <span class="math-container">$t=\sin(x)$</span>, <span class="math-container">$t=\tan(x)$</span>, and after expressing <span class="... | Ryszard Szwarc | 715,896 | <p>We have
<span class="math-container">$$(\sin x+2\cos x)(2\cos x-\sin x)=4-5\sin^2x=5\cos^2x-1$$</span>
Therefore
<span class="math-container">$${\sin^2x\over \sin x+2\cos x} ={\sin^2x(2\cos x-\sin x)\over (\sin x+2\cos x)(2\cos x-\sin x)}\\ =
2{\sin^2x\over 4-5\sin^2x}\cos x-{1-\cos^2x\over 5\cos^2x -1}\sin x$$</spa... |
986,412 | <p>Let $f(x) = \frac1{(1-x)}$.</p>
<p>Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$.</p>
<p>Find $f^{653}(56)$.</p>
<p><strong>What I've done:</strong> </p>
<p>I started with r=1,2,3 and noticed the following pattern:
$$f^r(x)= \left\{
\begin{array}{c}
\frac1{1-x}, when \ r\equiv 1\pmod 3 \\
\fr... | user84413 | 84,413 | <p>By the Fundamental Theorem of Calculus,</p>
<p>$\displaystyle f^{\prime}(\theta)=\frac{-\sin\theta}{1-\cos^{2}\theta}-\frac{\cos\theta}{\cos^{2}\theta}=-\csc\theta-\sec\theta$,</p>
<p>so $\displaystyle f(\theta)=-\left[\ln(\csc\theta-\cot\theta)+\ln(\sec\theta+\tan\theta)\right]+C=-\left[\ln((\csc\theta-\cot\theta... |
2,913,921 | <p>As I know, the general complexity of matrix inversion is $O(n^3)$,
but it is a little bit high.
My matrix is $(I + A)$ , where $I$ is an $n \times n$ identity matrix and $A$ is a hermitian matrix and the norm of all its elements are very small (near $1/100$). Therefore, I think it's a particular matrix that close... | Community | -1 | <p>There is something called a <a href="https://www.sciencedirect.com/science/article/pii/S1063520307001364" rel="nofollow noreferrer">fast randomized low-rank SVD</a>. The time complexity is $\mathcal{O}(n^{2}\log(k) +nl^{2}) $ The inversion at the end doesn't take much time. There are other similar matrix decomposi... |
652,025 | <p>Assume $A$, $B$, and $C$ are three independent predicates. Maybe $A$ stands for "my age is 20," and $B$ "stands for tomorrow is a good day."</p>
<p>So is it true that $(A \lor B) \land C \iff (A \lor C) \land (B \lor C)$?</p>
| amWhy | 9,003 | <p>No, it's NOT true that $$(A \lor B) \land C \equiv (A \lor C) \land (B \lor C)\tag{NOT TRUE}$$</p>
<p>What is true, by the distributive law, is that $$(A\lor B)\land C \equiv (A\land C)\lor (B\land C)\tag{TRUE}$$</p>
<p>Confirm the correct equivalence below:</p>
<p><img src="https://i.stack.imgur.com/bqdET.gif" a... |
640,769 | <p>I'm to prove that every proper ideal is a product of maximal ideals which are uniquely determined up to order.
I have no idea even how to start in the proof to solve this question :(
May anybody help me ? </p>
| Siminore | 29,672 | <p>Write your assumption as $F'(x) \leq c F(x)$. You know that $F(0)=0$. Fix $x_0>0$ and define
$$
M_0 = \sup_{x \in [0,x_0]} F(x), \quad M_1 =\sup_{x \in [0,x_0]} F'(x).
$$
Then, by the Mean Value Theorem, for every $x \in [0,x_0]$, $F(x) \leq M_1 x_0 \leq c x_0 M_0$. If $c x_0 < 1$, then $F(x)=0$ for every $x \... |
1,776,260 | <p>After understanding the Cardano's formula for solving the depressed cubic (of the form $x^3+mx=n$, of course), I tried to find the solution of the equation $$x^3+6x=20.$$
After plugging into the formula
$$x=(n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}+(-n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}$$
where... | DeepSea | 101,504 | <p>Let $a^3 = 10+\sqrt{108}=10+6\sqrt{3}, b^3 = -10 + \sqrt{108}=-10+6\sqrt{3}\implies a^3-b^3 = 20 = (a-b)(a^2+ab+b^2)=(a-b)((a-b)^2 + 3ab)$. Observe that $(ab)^3 = 8 \implies ab = 2\implies (a-b)((a-b)^2 + 6) = 20\implies x^3+6x-20 = 0, x = a-b$. Using synthetic division we have: $x^3+6x-20 = 0\implies (x-2)((x+1)^2 ... |
4,602,596 | <p>In an exercise, my teacher asked us to prove that <span class="math-container">$\ell^1$</span> is a Banach space. I was able to do so, but there are two steps in my proof that I'm not quite so sure they are correct. This is what I came up with:</p>
<hr />
<p>let <span class="math-container">$x^n=(x^n_1, x^n_2, ...)$... | Gerd | 960,044 | <p>If <span class="math-container">$f$</span> is bounded (<span class="math-container">$|f(t)| \le b$</span>) then
<span class="math-container">$$
y(t)=e^{-t} \int_{-\infty}^t e^\tau f(\tau)d\tau
$$</span>
is a bounded solution. Note that the integral is convergent since <span class="math-container">$f$</span> is bound... |
1,561,563 | <p>Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents to $\Gamma_1$ at $C$ and to $\Gamma_2$ at $D$ intersect at $M$. Let $N=AM\cap CD$. Let $l$ be a line through $N$ paral... | Ron Gordon | 53,268 | <p>There's a simple solution somewhere in M.SE which goes like this...</p>
<p>$$I = \int_0^{\pi/2} dt \frac{\cos{t}}{\cos{t}+\sin{t}} $$
$$J = \int_0^{\pi/2} dt \frac{\sin{t}}{\cos{t}+\sin{t}} $$</p>
<p>$$I + J = \frac{\pi}{2}$$
$$I - J = \int_0^{\pi/2} dt \frac{\cos{t}-\sin{t}}{\cos{t}+\sin{t}} = \int_0^{\pi/2} dt \... |
1,561,563 | <p>Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents to $\Gamma_1$ at $C$ and to $\Gamma_2$ at $D$ intersect at $M$. Let $N=AM\cap CD$. Let $l$ be a line through $N$ paral... | Claude Leibovici | 82,404 | <p>Since no one used it, let me consider the case of the antiderivative $$I=\int\frac{\cos x}{\sin x + \cos x} \space dx$$ Now, use the tangent half-angle substitution (Weierstrass substitution) $t=\tan(\frac x 2)$.</p>
<p>We so obtain $$I=\int \frac{2 \left(t^2-1\right)}{t^4-2 t^3-2 t-1}\space dt$$ But $$t^4-2 t^3-2 ... |
2,899,829 | <p>$\newcommand{\d}{\mathrm{d}}$</p>
<blockquote>
<p>Evaluate the integral using the indicated substituion. $$\int \cot x \csc^2x \,\d{x}, \qquad u= \cot x .$$</p>
</blockquote>
<p>Differentiating both sides of $u$, then making the substitution: $$
\begin{align}
u &=
\phantom{-}\cot x, \\
\d u &= -\cot x\c... | Entrepreneur | 571,353 | <p><strong>For alternative way:</strong></p>
<p>$$\int \cot x \csc^2 x dx$$
$$=\int \frac{\cos x dx}{\sin^3 x}$$</p>
<p>Now you can advance taking $\sin x = z$ .</p>
|
1,499,949 | <p>Prove that for all event $A,B$</p>
<p>$P(A\cap B)+P(A\cap \bar B)=P(A)$</p>
<p><strong>My attempt:</strong></p>
<p>Formula: $\color{blue}{P(A\cap B)=P(A)+P(B)-P(A\cup B)}$</p>
<p>$=\overbrace {P(A)+P(B)-P(A\cup B)}^{=P(A\cap B)}+\overbrace {P(A)+P(\bar B)-P(A\cup \bar B}^{=P(A\cap \bar B)})$</p>
<p>$=2P(A)+\un... | Koro | 266,435 | <p>Yes , there is a slight difference as per one of the conventions. <br>
Consider, $x^{1/2}$ and $\sqrt x$, former is written in a general sense(more appropriate to use) but the latter one is used when $x\ge 0$.So according to this $i^{1/2}$ and not $\sqrt i$.<br>
Hope this helps(not fully but to some extent atleast... |
596,005 | <p>Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation
$$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$
for all points $(x,y) \in \mathbb{R}^2$ if and only if for all $(x,y)\in \mathbb{R}^2$ and $t \in \mathbb{R}$ we have:
$$f(x, y + 2t) + f(x, y) = f(x + t,y + t) ... | Yiorgos S. Smyrlis | 57,021 | <p>First assume that (replacing $y$ by $y-t$ in the assumption)
$$
f(x, y + t) + f(x, y) = f(x + t,y) + f(x - t, y -t),
$$
for all $t,x,y\in\mathbb R$. This implies that
$$
\frac{f(x, y + t) -2f(x,y)+ f(x, y-t)}{t^2} = \frac{f(x + t,y)-2f(x,y) + f(x - t, y)}{t^2},
$$
for $x,y\in\mathbb R$ and $t\ne 0$. We shall now exp... |
2,998,189 | <p>I'm looking at a matrix operator in which <span class="math-container">$T \in \mathcal{L}(\mathbb{R}^2)$</span> by <span class="math-container">$T(x,y) = (x, -y)$</span>. So its basis is <span class="math-container">$ \mathcal{M}(T) = \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$</span>. </p>
<p>How do I show ... | lhf | 589 | <p><em>Hint:</em> An operator is self-adjoint iff its matrix is self-adjoint. The adjoint of a matrix is its conjugate transpose.</p>
|
3,371,638 | <p>Measure space <span class="math-container">$(X, \mathcal{A}, ν)$</span> has <span class="math-container">$ν(X) = 1$</span>. Let <span class="math-container">$A_n \in \mathcal{A} $</span> and denote </p>
<p><span class="math-container">$B := \{x : x ∈ A_n$</span> for infinitly many n }.</p>
<p>I want to prove that... | Lee Mosher | 26,501 | <p>Let me use the standard terminology <em>disc</em> to refer to a circle union its interior. Also, let me use the standard terminology <em>polygon</em> to refer to a polyhedron in the plane.</p>
<p>Each circle is the boundary of its corresponding disc, and that boundary has the following special relation with the cen... |
41,174 | <p>I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a refere... | Dan Fox | 9,471 | <p>See Ó. Garcia-Prada's appendix to the third edition of R.O. Wells' book Differential Analysis on Complex Manifolds. It addresses most of what you are asking quite explicitly, in the context of Riemann surfaces, and has references to the original papers. Also the book of Lübke and Teleman on the Kobayashi-Hitchin cor... |
41,174 | <p>I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a refere... | Chandan Singh Dalawat | 2,821 | <p>I'm surprised nobody has mentioned Le Potier's Bourbaki exposé :</p>
<p><a href="http://www.numdam.org/item?id=SB_1990-1991__33__221_0">Fibrés de Higgs et systèmes locaux</a>, Séminaire Bourbaki, 33 (1990-1991), Exposé No. 737, 48 p. </p>
|
2,088,229 | <p>Is there a neat way to find the largest integer that divides another integer fully, within a range. As an example, I would like to find the largest integer smaller than 1131 that divides 3500 completely. </p>
<p>So far I have just tried by breaking up 3500 into its prime components and guessing, coming to 875, but ... | Henry | 6,460 | <p>You could look for the smallest integer greater than $\dfrac{3500}{1131}\approx 3.09$ that divides $3500$ exactly </p>
<p>This is obviously $4$ (there is no smaller integer greater then $3.09$ and $4$ does divide $3500$ as it divides $100$) so the answer to your original question is $\dfrac{3500}{4}=875$</p>
|
148,127 | <p>I am trying to get an analytic expression for this integral:</p>
<pre><code> Integrate[Sign[Cos[q]]/(q + 1), {q, 0, x}, Assumptions -> x > 0]
</code></pre>
<p>Mathematica gives the answer:</p>
<pre><code> Abs[Cos[x]] Log[1 + x] Sec[x]
</code></pre>
<p>However, when I compare it to the numerical integration... | Alan | 19,530 | <p>Use <code>Labeled</code>.</p>
<pre><code>f = Function[{x, y, z}, x z + y z - x y z]
tmp = StringTemplate["C=``"]
plots = Labeled[
ContourPlot3D[f[x, y, z] == #, {x, 0, 2}, {y, 0, 2}, {z, 0, 2}],
tmp[#]] & /@ {0.0, 0.2, 0.4};
Grid[{plots}]
</code></pre>
<p><a href="https://i.stack.imgur.com/3y8ue.png"... |
3,975,832 | <p>I think the following claim is clearly correct, but I cannot prove it.</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be sets. If <span class="math-container">$f:A \times B \to \mathbb{R}$</span> satisfies <span class="math-container">$f(a, b) \leq C_a$</s... | Kavi Rama Murthy | 142,385 | <p>Even an upper bound need not exist. Take <span class="math-container">$f:\mathbb R \times \mathbb R \to \mathbb R$</span> defined by <span class="math-container">$f(x,y)=x$</span> if <span class="math-container">$x=y$</span> and <span class="math-container">$0$</span> if <span class="math-container">$x \neq y$</span... |
209,842 | <p>I was looking for a general way of formulating solutions for work and time problems.</p>
<p>For example,</p>
<p>30 soldiers can dig 10 trenches of size 8*3*3 ft in half a day working 8 hours per day. How many hours will 20 soldiers take to dig 18 trenches of size 6*2*2 ft working 10 hours per day?</p>
<p>Now i kn... | Brian M. Scott | 12,042 | <p>Identify the basic assumptions. In this problem it’s clear that soldiers are considered interchangeable: they all work at the same rate. (Recall that in some problems we have different workers working at different rates and have to keep track of the work rate of each worker. And it’s the work rates that are importan... |
840,700 | <p>I have to show that the following function $f:[0,1]\rightarrow\mathbb{R}$ is Riemann Integrable:</p>
<p>$$f(x) =
\left\{
\begin{array}{ll}
1 & \mbox{if } x = \frac{1}{n} \\
0 & \mbox{otherwise}
\end{array}
\right.$$</p>
<p>For the upper and lower Riemann sum I am using the following definitions:</p>... | Badoe | 97,642 | <p>$ou have the right idea. It's all about the technicalities. One possible partition set-up is as follows.</p>
<p>For every $\epsilon>0$, there exists an integer $N_0$ such that $1/N_0<\epsilon$. Let $N=\max\{N_0,5\}$.</p>
<p>Partition $[0,1]$ with $V=\{0,x_1, x_2,\ldots,x_{4N-5}\}$ where $x_1=\frac{1}{2N}$, $... |
4,124,777 | <p>I'm trying to find out whether <span class="math-container">$\sum _{n=0}^{\infty }\left(\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}\right)$</span> converges or not. I've tried with taylor series but it doesn't lead me anywhere except with the fact that <span class="math-container">$\lim_{n \to \infty}\c... | Riemann'sPointyNose | 794,524 | <p><strong>EDIT</strong>: I decided to edit my answer to make it more explicit.</p>
<p>To straight up answer the question: is there a difference between <span class="math-container">${\mathbb{R}}$</span> and <span class="math-container">${\mathbb{R}^1}$</span>? Yes, technically there is a difference.</p>
<p>When we spe... |
1,177,721 | <p>A fair $6$-sided die is rolled $6$ times independently. For any outcome, this is the set of numbers that showed up at least once in the different rolls. For example, the outcome is $(2,3,3,3,5,5)$, the element set is $\{2,3,5\}$. What is the probability the element set has exactly $2$ elements? how about $3$ element... | Jeny | 221,588 | <p>Why can't we use the multinomial formula? We want to have a sequence of 2 repeated elements 3 each. This sounds like $\frac{6!}{3!*3!}$, right?
For example we have {1,1,1,2,2,2} or a combination of this. Then we get $\frac{6!}{3!*3!}$ since we have 2 sets of 3 repeated elements and then we divide by the sample spac... |
3,091,353 | <p>There are 2 definitions of <strong><em>Connected Space</em></strong> in my lecture notes, I understand the first one but not the second. The first one is:</p>
<blockquote>
<p>A topological space <span class="math-container">$(X,\mathcal{T})$</span> is connected if there does not exist
<span class="math-conta... | Chris Culter | 87,023 | <p>Hint: Consider the following definition, halfway between the first and second:</p>
<p><span class="math-container">$$\mbox{A topological space } (X,\mathcal{T}) \mbox{ is connected if there does not exist } U\in\mathcal{T} \mbox{ such that } U\neq\emptyset, (X\setminus U)\neq\emptyset, (X\setminus U)\in\mathcal{T} ... |
1,992,143 | <p>I'm trying to determine if $\sum \limits_{n=1}^{\infty} \sin(n\pi + \frac{1}{2n})$ absolutely converges or not.</p>
<p>Help me check it. I don't know how to do it. Advance thanks. :)</p>
| hamam_Abdallah | 369,188 | <p>For each $n>0$</p>
<p>$\sin(n\pi+\frac{1}{2n})=(-1)^n\sin(\frac{1}{2n})$</p>
<p>and</p>
<p>$$|\sin(n\pi+\frac{1}{2n})|=\sin(\frac{1}{2n}).$$</p>
<p>but</p>
<p>$$\sin(\frac{1}{2n})=\frac{1}{2n}(1+(2n\sin(\frac{1}{2n})-1))$$</p>
<p>$$=\frac{1}{2n}(1+\epsilon(n))$$ with</p>
<p>$$\lim_{n\to+\infty}\epsilon(n)=... |
1,992,143 | <p>I'm trying to determine if $\sum \limits_{n=1}^{\infty} \sin(n\pi + \frac{1}{2n})$ absolutely converges or not.</p>
<p>Help me check it. I don't know how to do it. Advance thanks. :)</p>
| Astyx | 377,528 | <p>Notice that $|\sin(n\pi +{1\over2n})| = \sin({1\over2n}) \sim {1\over2n}$ and therefore $\sum \sin(n\pi +{1\over2n})$ is absolutely divergent (because $\sum{1\over n}$ is)</p>
|
3,601,865 | <p>What does the correspondence theorem (or 4th isomorphism theorem for rings) for rings mean and how is it used? That is, why do we care about it?</p>
<hr>
<p>Edit:</p>
<p>My version of the correspondence theorem:</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a ring and <span class="math-con... | Alex Kruckman | 7,062 | <p>One of the most basic applications of this theorem is the result that a quotient ring <span class="math-container">$R/I$</span> is a field if and only if <span class="math-container">$I$</span> is a maximal ideal. (For me, all rings are commutative with unity.) </p>
<p>Every non-zero ring <span class="math-containe... |
3,717,506 | <p>I am reading some text about even functions and found this snippet:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be an integrable even function. Then,</p>
<p><span class="math-container">$$\int_{-a}^0f(x)dx = \int_0^af(x)dx, \forall a \in \mathbb{R}$$</span></p>
<p>and therefore,</p>
<p><span c... | Ross Millikan | 1,827 | <p>The limit that you hit any number <span class="math-container">$k$</span>, positive or negative, goes to <span class="math-container">$1$</span> as <span class="math-container">$n \to \infty$</span>. Say we want the chance of hitting <span class="math-container">$k=27$</span>. This is higher than the chance we get... |
4,043,787 | <p>I have <span class="math-container">$$f_n(x)=\begin{cases}
\frac{1}{n} & |x|\leq n, \\
0 & |x|>n .
\end{cases}$$</span></p>
<p>Why cannot be dominated by an integrable function <span class="math-container">$g$</span> by the Dominated Convergence Theorem? I am also wondering what exactly it... | RobertTheTutor | 883,326 | <p>In order to dominate a sequence of functions, it has to be larger than ALL of the functions in the entire sequence. There is no such <span class="math-container">$g$</span> for your sequence of functions, because no matter how large you set <span class="math-container">$g$</span> near <span class="math-container">$... |
4,498,498 | <p>Lets say you have two sequences of non negative integers each of length <span class="math-container">$n$</span>.</p>
<p>ie <span class="math-container">$(a_1,a_2,...,a_n)$</span> and <span class="math-container">$(b_1,b_2,...,b_n)$</span> such that
<span class="math-container">$\max(a_i) < k$</span>
and
<span cla... | ploosu2 | 111,594 | <p><em>Disclaimer:</em> This is just a computational answer but it gives a formula for at least for the case <span class="math-container">$k=2$</span> (and other small cases too). And it's too long for a comment.</p>
<p>I get these values for <span class="math-container">$f(k, n) = $</span> the number of these solvable... |
374,380 | <p>I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not. Here's what I am thinking.</p>
<p>Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup $N=(\mathbb{Z},+)$. Then I form $G/N$. So this thing identifies any real number $x$ with the integers that are exact... | Mikasa | 8,581 | <p>You can also use the following nice facts. I hope you are inspired by them.</p>
<p>$$\mathbb R/\mathbb Z\cong T\cong\prod_p\mathbb Z(p^{\infty})\cong\mathbb R\oplus(\mathbb Q/\mathbb Z)\cong\mathbb C^{\times}$$</p>
|
374,380 | <p>I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not. Here's what I am thinking.</p>
<p>Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup $N=(\mathbb{Z},+)$. Then I form $G/N$. So this thing identifies any real number $x$ with the integers that are exact... | Xi Wu | 24,367 | <p>Just try to add some intuitive answers:</p>
<ol>
<li><p>The essential intuition that <span class="math-container">$\mathbb{R}/\mathbb{Z}$</span> is isomorphic to unit circle in the complex plane is that we can map a real number <span class="math-container">$x$</span> to the radian so we get <span class="math-contain... |
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | Martin Argerami | 22,857 | <p>The matrix is a unitary conjugate (via permutations) of
$$\begin{bmatrix}1&1&0&0\\1&1&0&0\\0&0&1&1\\0&0&1&1\end{bmatrix} $$(concretely, use the permutation corresponding to the exchange of the second and third elements of the canonical basis). So the characteristic p... |
2,231,003 | <p>I am trying to prove that the function $f_a(x) = e^{-\frac{1}{(x-a)}}$ is differentiable for all $x>a$. However, I do not know how to show $|\frac{e^{-\frac{1}{(x-a)}}-e^{-\frac{1}{(p-a)}}}{x-p} - (-\frac{e^{-\frac{1}{(p-a)}}}{(p-a)^2})| < \epsilon$ if $0<|x-p|<\delta$. </p>
<p>My next idea was to apply... | marshal craft | 167,793 | <p>There are plenty of proofs for the chain rule, that is if $f(x)$ and $g(x)$ are differentiable, then so is $g(f(x))$. You can take it upon yourself to verify this, as it is typically standard for students to prove these statements they use every day in calculus. As such you can find proofs of such, I believe I have ... |
241,903 | <p>Suppose $f: \mathbb{D}\to \mathbb{C}$ is a univalent function with $$f(z)=z+a_2z^2+a_3z^3+\cdots.$$ The Bieberbach conjecture/de Branges' theorem asserts that $|a_n|\leq n$ with equality for the Koebe function, which has an unbounded image. Suppose we restrict to the class of univalent functions whose image is actua... | Alexandre Eremenko | 25,510 | <p>The answer is yes, even the rate of increase of coefficients is much smaller. However even the precise rate is not known (not speaking of the exact estimate).
The strongest results in this direction are here:</p>
<p>MR1162188
Carleson, Lennart; Jones, Peter W.
On coefficient problems for univalent functions and co... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Tom Leinster | 586 | <p><em>Introduction to Higher Order Categorical Logic</em> by Lambek and Scott might fit the bill. </p>
|
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Paul Taylor | 2,733 | <p>Well, for the record, there is also my
<a href="http://paultaylor.eu/prafm/"><i>Practical Foundations of Mathematics</i></a> (CUP, 1999).</p>
|
3,063,742 | <p>Consider the closed interval <span class="math-container">$[0, 1]$</span> in the real line <span class="math-container">$\mathbb{R}$</span> and the product space <span class="math-container">$([0, 1]^{\mathbb{N}}, τ ),$</span></p>
<p>where <span class="math-container">$τ$</span> is a topology on <span class="math-c... | sranthrop | 95,054 | <p>Note that both <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are closed in <span class="math-container">$\mathbb R^2$</span> (why?), and so <span class="math-container">$M:=\mathbb R^2\backslash B$</span> and <span class="math-container">$N:=\mathbb R^2\backslash A$</span> are o... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| Jan Eerland | 226,665 | <p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}=$$
$$\sqrt[3]{8+12\sqrt{5}+30+5\sqrt{5}} - \sqrt[3]{-8+12\sqrt{5}-30+5\sqrt{5}}=$$
$$\sqrt[3]{8+12\sqrt{5}+6\left(\sqrt{5}\right)^2+\left(\sqrt{5}\right)^3} - \sqrt[3]{-8+12\sqrt{5}-6\left(\sqrt{5}\right)^2+\left(\sqrt{5}\right)^3}=$$
$$\sqrt[3]{\left(\sqrt{5}... |
1,866,931 | <p>I would like to see a proof to this fact.</p>
<blockquote>
<p>If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there holds
$$
\|B-A\| \|A^{-1}\| <1,
$$
we have that B is invertible.</p>
</blockquote>
<p>Moreov... | be5tan | 339,513 | <p>I think you have to start with with given elements $x, y, z \in A: xRy, yRz$. Now you have to show that $xRz$. However you started with an elements $x, z\in A: (x, z) \in R\circ R$, which is the wrong antecedent for the implication you want to prove.</p>
<p>BS</p>
|
755,571 | <p>$$a_n=3a_{n-1}+1; a_0=1$$</p>
<p>The book has the answer as: $$\frac{3^{n+1}-1}{2}$$</p>
<p>However, I have the answer as: $$\frac{3^{n}-1}{2}$$</p>
<p>Based on:</p>
<p><img src="https://i.stack.imgur.com/4vJrQ.png" alt="enter image description here"></p>
<p>Which one is correct?</p>
<p>Using backwards substit... | David | 119,775 | <p>First note that "backwards substitution" is not a great way to solve this sort of thing, there are much better methods which I assume you will learn at some stage in your course.</p>
<p>However if you want to do it this way you can, only your working is not accurate. Try this:
$$\eqalign{a_n
&=3a_{n-1}+1\cr
... |
1,268,598 | <p>I thought the <code>|</code> symbol meant "divides by", but in set theory it seems that it means "such that." However, I thought we wrote "such that" as <code>:</code>.</p>
<p>Can anybody elaborate?</p>
| Community | -1 | <p>Any of the following might mean "such that":</p>
<p>$: \ ; \ , \ / \ | \ s.t.$ </p>
<p>I don't know if there are others.</p>
|
1,268,598 | <p>I thought the <code>|</code> symbol meant "divides by", but in set theory it seems that it means "such that." However, I thought we wrote "such that" as <code>:</code>.</p>
<p>Can anybody elaborate?</p>
| Asaf Karagila | 622 | <p>This shouldn't be surprising. There are only so many letters that we can easily access.</p>
<ol>
<li><p>$\Bbb{P,Q,R}$ can denote a forcing notion, and have nothing to do with probability, the rationals or the reals. </p></li>
<li><p>$\pi$ denotes a real number, the ratio between a diameter and circumference of a ci... |
949,664 | <p>Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$ <br/>
Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$. <br/>
Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$.<br/>
The question is, what will happen to the $f(x)$ after getting transformed?<br/>
Why should the function... | Paul Sundheim | 88,038 | <p>One major advantage lies in the fact that many differential equations become algebraic equations when the Laplace transform is applied. We can then solve the algebraic equations and take the inverse Laplace transform (the transform is one to one and so has an inverse) to arrive at a solution to the differential equ... |
1,955,393 | <p>I have been trying to evaluate this limit:</p>
<p>$$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}$$</p>
<p>What methods should I try in order to proceed?</p>
<p>I was advised to use "Limit Chain Rule", but I believe there is a different approach.</p>
| Simply Beautiful Art | 272,831 | <p>By the root test for a power series, this equivalent to the reciprocal of the radius of convergence of the following:</p>
<p>$$\sum_{n=0}^\infty(4^n+5^n)x^n$$</p>
<p>Which is a basic geometric series with radius of convergence $1/5$, so the limit is</p>
<p>$$\lim_{n\to\infty}\sqrt[n]{4^n+5^n}=1/R=5$$</p>
|
11,609 | <p>What's the code for multiple alignment in MathJAX? An analogous question for Latex is at <a href="https://tex.stackexchange.com/questions/43464/multiple-alignment-in-equations">https://tex.stackexchange.com/questions/43464/multiple-alignment-in-equations</a>, but it doesn't appear to function here. Thank you.</p>
<... | Zev Chonoles | 264 | <p>This is pretty hack-ish, but </p>
<pre>
$$\newcommand{\myeq}{\!\!=\!\!}
\begin{array}{ccccccccc}
A & \myeq & B \\
& & B & \myeq & C \\
& & & & C & \myeq & D \\
& ... |
3,454,146 | <p>I'm trying to wrap my head around this new subject. I have to determine the validity of this argument (using a truth table): </p>
<p>"If Steve went to the movies then Maria's sister would not have stayed home. Either Steve went to the movies or Maria or both. If Maria went, then Maria’s sister would have st... | RyRy the Fly Guy | 412,727 | <p>Your translation into propositional logic looks great! Note that the inclusive disjunction <span class="math-container">$\vee$</span> is true whenever one or both operands are true, so the statement "either Steve went to the movies or Maria or both" may be written as <span class="math-container">$P \vee R$</span> as... |
898,543 | <p>I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be simply the integral $\int_{0}^{\sqrt{x}}8x^{2}y = 4x^{3}$, which is also the given solution, and follows the general... | Umberto P. | 67,536 | <p>Use $P(A | B) = P(A \cap B)/P(B)$. You are given $P(L|B) = \frac 17$ and $P(B|L) = \frac 13$. Thus $P(B) = 7 P(B \cap L)$ and $P(L) = 3 P(B \cap L)$.</p>
<p>You are also given $P(B \cup L) = \frac 15$ since this is the probability of having either attribute. Since $P(B \cup L) + P(B \cap L) = P(B) + P(L)$ you obtai... |
200,876 | <p>Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?</p>
<blockquote>
<blockquote>
<p>A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) = y$.</p>
</blockquote>
</bloc... | Helene Sigloch | 62,434 | <p>First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality... |
2,130,776 | <p>Let $(L,R)$ be a pair of adjoint functor. </p>
<p>How to show that the commutativity of the left diagram induces the commutativity of the right one?</p>
<p><a href="https://i.stack.imgur.com/lOffl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lOffl.jpg" alt="diagram"></a></p>
| Community | -1 | <p>Suppose $L : \mathcal{C} \to \mathcal{D}$ is left adjoint to $R : \mathcal{D} \to \mathcal{C}$, with unit $\eta : 1_\mathcal{C} \to RL$ and counit $\epsilon : LR \to 1_\mathcal{D}$.</p>
<p>We can prove the following, more general statement: let $J$ be a category, and let $[J, X]$ be the category of functors from $J... |
2,087,084 | <p>Fix an arbitrary abelian category $\mathscr{A}$, and let $$0\to A\xrightarrow{f}B\xrightarrow{g}C\to 0$$ be a short exact sequence in the category of chains $\mathscr{A}_\bullet$, where $A$, $B$, and $C$ have chain maps $\varphi^A_n:A_n\to A_{n-1}$, $\varphi^B_n:B_n\to B_{n-1}$, $\varphi^C_n:C_n\to C_{n-1}$ respecti... | Pedro | 23,350 | <p>The <em>long exact homology sequence</em> is <strong>the</strong> result to prove that if two out of three complexes in a SEC are exact, so is the third. Prove this, and you're done. To do so, if suffices you prove the well known <em>snake lemma</em> for an arbitrary abelian category. </p>
|
2,762,953 | <p>I've studied Markov Process with 2x2 matrices.
Using the linear algebra and calculus procedures is clear to me how a Markov chain works.</p>
<p>However, i'm still not able to grasp the intuitive and immediate meaning of a Markov chain.
Why intuitively, for $n\rightarrow +\infty $, the state of the system is indepen... | Siong Thye Goh | 306,553 | <p>It does depends on the initial state. Consider the Markov chain which consist of two disjoint state, that is $$\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$$</p>
<p>Then if you start in state $1$, you will stuck in state $1$ forever as it is an absorbing state. Similarly if you start in state $2$. </p>
<p>... |
4,357,484 | <p>Suppose the following series:
<span class="math-container">\begin{eqnarray}
\sum_{k'}k'f_{k'}
\end{eqnarray}</span>
where <span class="math-container">$f_{k'}$</span> are some Fourier coefficients that result from a periodic function <span class="math-container">$f(t+T)=f(t)$</span>:
<span class="math-container">\be... | Captain Lama | 318,467 | <p>To provide an alternative (faster) proof than the method suggested by tomasz, you may note that <span class="math-container">$A$</span> being integral means that for any <span class="math-container">$a\neq 0$</span>, the application <span class="math-container">$A\to A$</span> defined <span class="math-container">$x... |
3,394,378 | <p>I am stuck with this Precalculus problem about polynomial functions. The problem:</p>
<blockquote>
<p>Consider <span class="math-container">$f(x)=x^2+ax+b$</span> with <span class="math-container">$a^2-4b>0$</span>. Let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> b... | October15 | 715,022 | <p>By parallely translating the graph of the given <span class="math-container">$f(x)$</span>, we can get the graph of any <span class="math-container">$g(x)=x^2+cx+d$</span>. So it is sufficient to find the minimum of any <span class="math-container">$g(x)$</span> whose solutions is 2 apart: for example, <span class="... |
281,504 | <p>Is there a trick for easily solving a matrix polynomial like
$$
p(A) = \left( 7\cdot A^4 - 4\cdot A^3 + 6\cdot A - 5\cdot E \right)
, A = \left(\begin{matrix}2 & -1 \\ 3 & 5\end{matrix}\right)
$$
or is it really just step-by-step calculation of
$$
7\cdot A\cdot A\cdot A\cdot A\cdot A\cdot A\cdot A-4\cdot A\c... | Ross Millikan | 1,827 | <p>Yes, if by reflect you mean to draw a perpendicular from the point to the line and continue it the same distance on the other side. If your line is $(p_x,p_y,p_z)+t(q_x,q_y,q_z)$ (is this what you mean by vector equation?) and the point is $(r_x,r_y,r_z)$ the point on the line where the perpendicular hits can be fo... |
2,069,573 | <p>I'm trying to solve the following inequality $\dfrac{(\log_2 (8x) \times \log_{x/8} 2)}{\log_{x/2} 16} \leq 0.25$ </p>
<p>Wolfram alpha gives the answer $(0, 0.5], [1,8)$ but surely $x \not= 2$ since log to base $1$ is undefined. But is the fact that it basically shrinks the fraction down to $0$ sufficient to satis... | Arnaldo | 391,612 | <p>Put everything on the base $2$:</p>
<p>$$\dfrac{(\log_2 (8x) \times \log_{x/8} 2)}{\log_{x/2} 16} \leq 0.25$$</p>
<p>$$\dfrac{\log_2 (8x) }{\log_{x/2} 16}\times \frac{\log_{2} 2}{\log_{2} (x/8)} \leq 0.25$$</p>
<p>$$\log_2(8x)\times\dfrac{\log_2 (x/2) }{\log_{2} 16}\times \frac{\log_{2} 2}{\log_{2} (x/8)} \leq 0.... |
1,448,363 | <p>I have gotten to the next stage where you write it as $\frac{1}{\left(\frac 34\right)}$ to the power of $3$, now I am stuck</p>
<p>I've got it now, thanks everyone.</p>
| Mark | 270,016 | <p>$$=\left(\frac{3}{4}\right)^{-3}=\left(\frac{1}{\frac{3}{4}}\right)^3=\left(\frac{4}{3}\right)^3=\frac{4^3}{3^3}=\frac{64}{27}\approx2.37$$</p>
|
2,491,818 | <p>My professor showed us that the Cauchy distribution does not have an expected value, that is, the integral $\int_{-\infty}^{\infty} x f(x) \text{d} x$ is not defined ($f(x)$ is the p.d.f. of the Cauchy distribution). I find that very counterintuitive. What does it actually mean, in the context of probability, to not... | Math1000 | 38,584 | <p>For a random variable $X$, write $X=X^+-X^-$, where $X^+=\max\{X,0\}$ and $X^-=\max\{-X,0\}$. Then $|X|=X^++X^-$, so $\mathbb E[|X|]<\infty$ if and only if $\mathbb E[X^+]<\infty$ and $\mathbb E[X^-]<\infty$. When this condition holds, we define
$$
\mathbb E[X] = \mathbb E[X^+-X^-]= \mathbb E[X^+]-\mathbb E... |
2,617,235 | <p>Given a triangle $\Delta$ABC, how to draw any inscribed equilateral triangle whose vertices lie on different sides of $\Delta$ABC?</p>
| Alex | 293,781 | <p><strong>Construction</strong></p>
<ol>
<li><p>Use <a href="https://en.wikipedia.org/wiki/Morley%27s_trisector_theorem" rel="nofollow noreferrer">Morley's theorem</a> to construct an equilateral triangle $\triangle XYZ$ that lies strictly inside $\triangle ABC$.</p></li>
<li><p>Through the vertices of $\triangle XYZ... |
1,242,760 | <p>Now proving by induction is fairly simple. However, this is a multiple choice problem whose answers don't make any sense to me. The actual problem goes as follows:</p>
<p><em>To prove by induction that $n^2 - 7n - 2$ is divisible by $2$ is true for all positive integers $n$, we assume $k^2 - 7k - 2$ is divisible b... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ If $\,f(0)\,$ is even and $\,f(n\!+\!1)-f(n)=2j\,$ is even then by induction $\,f(n)\,$ is even for all $\,n,\,$ the inductive step being: $\ \,f(n+1) = f(n) +2j = $ even + even = even.</p>
<p><strong>Remark</strong> $\ $ mod $\,2\,$ it is simply: $ $ if $f\,$ is constant $\,f(n\!+\!1)\equ... |
1,003,379 | <p>I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate</p>
<p><img src="https://i.stack.imgur.com/HWRoM.jpg" alt="enter im... | Robert Israel | 8,508 | <p>Hint: $1/z$ has a pole at $z=0$.</p>
|
690,621 | <p>Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. </p>
<p>Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point to note here is $(5) \subset (x^2+3)$. So, we can consider $\mathbb{Z_5}[x]/(5)$. But this is just $\mathbb{Z... | Elchanan Solomon | 647 | <p>Let's start with an example. You have a table with an assortment of different items. You bring in three people: somebody with good eyesight, somebody with okay eyesight, and somebody who has terrible eyesight. You ask the three people to distinguish the items on the table. The person with good eyesight sees that eve... |
1,350,085 | <p>Consider $f(t)$, continuous on $[0,1]$, and $\alpha > 1$, and:</p>
<p>$$\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$$</p>
<p>How can we tell this integral diverges? Basically since $f$ is continuous it reaches it's minimum at $[0,1]$ so we could make a comparison with $\int_0^1 \frac{f(x_{min})}{t^{\alpha+1}}\ dt... | lulu | 252,071 | <p>Well, there are $2^{18}$ subsets all in all. Of these, $\dfrac{18!}{9!\cdot 9!}$ have exactly $9$ elements. The rest either have more than $9$ or fewer than $9$. Now, by symmetry, the number of subsets with more than $9$ elements equals the number of subsets with fewer than $9$ (pair each subset with its complem... |
1,791,837 | <p>Recently I have been very fascinated by the claim and impact of Godel's incompleteness theorem. To understand the proof given by Godel, I felt the need to read an introductory book in logic to begin with. I have started reading the book titled "A Mathematical Introduction to Logic" by Herbert Enderton. As mentioned ... | Thomas | 284,057 | <p>Write $S^n$ as the union of two open sets $U= S^n-N$, $V=S^n-S $where $N,S$ are the north and south pole. Note that the intersection $U\cap V$ is connected if $n\geq 2$. $U,V$ are contracile (diffeomorphic to $R^,$, and on each of these sets the form $\alpha$ is exact and andmits primitive $f_U, f_V$. On the inter... |
3,301,696 | <p>Prove that if <span class="math-container">$R$</span> is a non-commutative ring with <span class="math-container">$1$</span> and if <span class="math-container">$a,b \in R$</span> and if <span class="math-container">$ab =1 $</span> but <span class="math-container">$ba \neq 1$</span> then <span class="math-container... | lisyarus | 135,314 | <p>Consider the sequence of left ideals <span class="math-container">$R=Ra^0 \supset Ra^1 \supset Ra^2 \supset Ra^3 \supset\dots$</span> - we are going to show that all the inclusions are proper, meaning that there is in infinite number of ideals in <span class="math-container">$R$</span>, thus <span class="math-contai... |
362,716 | <p>Let <span class="math-container">$E$</span> be a separable <span class="math-container">$\mathbb R$</span>-Banach space, <span class="math-container">$\rho_r$</span> be a metric on <span class="math-container">$E$</span> for <span class="math-container">$r\in(0,1]$</span> with <span class="math-container">$\rho_r\le... | 0xbadf00d | 91,890 | <p>Building up on <a href="https://mathoverflow.net/a/362732/91890">Benoît Kloeckner's answer</a>, consider the following simplified scenerio: Let <span class="math-container">$(E,d)$</span> be a complete separable metric space, <span class="math-container">$(\kappa_t)_{\ge0}$</span> be a Markov semigroup on <span clas... |
1,497 | <p>If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-preserving functors from B to C) defined by precomposing with f.</p>
<p>Does anyone conditions on A,B,C under which this... | Tom Leinster | 586 | <p>I don't know the answer to that question, but I know the answer to some similar questions.</p>
<p>The simplest (which you probably know about) is that if f: A --> B is a map of (small) categories and C has all small colimits (respectively, limits) then f^* has a left (respectively, right) adjoint. These adjoints a... |
3,482,138 | <p><a href="https://i.stack.imgur.com/OUlV2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OUlV2.png" alt="enter image description here"></a></p>
<ol>
<li>I have found two planes trough the origin that meet the given plane
at right angles.</li>
</ol>
<p>I found three points in the plane, getting t... | dfnu | 480,425 | <p><strong>A fast approach using Euclidean geometry</strong></p>
<p><a href="https://i.stack.imgur.com/36iTB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/36iTB.png" alt="enter image description here"></a></p>
<ol>
<li><p>Since <span class="math-container">$\mathcal C_1$</span> and <span class="m... |
34,959 | <p>$F(x) = \int_{x-1}^{x+1}f(t)dt$ for x an element of the reals.</p>
<p>Show that $F$ is differentiable on Reals, and compute $F^\prime$.</p>
<p>I am unsure about how to showing $F$ is differentiable. I know that I need to use the fundamental theorem of calculus, but can someone please explain how to do so?</p>
| El Moro | 10,026 | <p>If you haven't done any measure theory , a simple answer would be: </p>
<p>If f is continuous then it has a primitive (the integral is supposed to be rieman one) If 0 belongs to the domain of f, let us then call G(x)=integral from 0 to x f(t)dt such a primitive function wich is differentiable (as f is its derivati... |
34,959 | <p>$F(x) = \int_{x-1}^{x+1}f(t)dt$ for x an element of the reals.</p>
<p>Show that $F$ is differentiable on Reals, and compute $F^\prime$.</p>
<p>I am unsure about how to showing $F$ is differentiable. I know that I need to use the fundamental theorem of calculus, but can someone please explain how to do so?</p>
| Aryabhata | 1,102 | <p>If $f$ is not continuous, then $F(x) = \int_{x-1}^{x+1} f(t) \text{d}t$ need not be differentiable at the points of discontinuity of $f$.</p>
<p>For instance I believe the following is a counter-example:</p>
<p>$ f(x) = \begin{cases} 0 & x \le 3 \\ 1 & x \gt 3 \end{cases}$</p>
<p>I believe we can show tha... |
637,061 | <p>I have a problem:</p>
<blockquote>
<p>For a system of linear equations:
$$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$
Prove that, if<br>
$$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$
then $(1)$ has a unique solution.</p>
</blockquote>
<p>==================================</p>
<p... | Ulrik | 53,012 | <p>Define $\pmb{A}$ to be the matrix with entries $a_{i,j}$ for $i,j = 1, \ldots, n$, $\pmb{x} = (x_1, \ldots, x_n)$ and $\pmb{b} = (b_1, \ldots, b_n)$. Then your system is equivalent to
$$\pmb{x} = \pmb{A}\pmb{x} + \pmb{b}$$
Define $f: \mathbb{R}^n \to \mathbb{R}^n$ by $f(\pmb{x}) = \pmb{A}\pmb{x} + \pmb{b}$. Equip $\... |
2,529,387 | <p><a href="https://i.stack.imgur.com/3M7iS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3M7iS.jpg" alt="enter image description here"></a></p>
<p>The fourth question is how can I do?</p>
<blockquote>
<p>Q: Given that $a, b \in \mathbb{Z}$ and $a$ is even, show that if $a^2b$ is not divisible ... | Mr. Brooks | 162,538 | <p>Is $a$ singly even or doubly even? If the former, that means $a = 4k + 2$, while the latter means $a = 4k$. If $a$ is singly even, then $a^2$ is doubly even, since $a^2 = (4k + 2)(4k + 2) = 16k^2 + 8k + 4$; however, $a^2$ is not divisible by $8$.</p>
<p>If $a$ is doubly even, then so is $a^2$, and what's more, it's... |
2,764,073 | <p>I recently was working on a question posted in an AP calculus BC multiple choice sheet which asked:</p>
<p>Let f(x) be a positive, continuous deceasing function. If $\int_1^∞ f(x)dx$ = 5, then which of the following statements must be true about the series $\sum_1^∞f(n)$?</p>
<p>(a) $\sum_1^∞f(n)$ = 0</p>
<p>(b) ... | Sri-Amirthan Theivendran | 302,692 | <p>Observe that
$$
\int_{1}^nf(x)\,dx=\sum_{i=1}^{n-1}\int_{i}^{i+1}f(x)\,dx<\sum_{i=1}^{n-1}\tag{1}
f(i)$$
for any integer $n\ge 1$ where we have using the fact that $f$ is decreasing for the inequality. Let $n\to\infty$ in (1) to get that
$$
5\leq\sum_{i=1}^\infty f(i)
$$</p>
|
2,764,073 | <p>I recently was working on a question posted in an AP calculus BC multiple choice sheet which asked:</p>
<p>Let f(x) be a positive, continuous deceasing function. If $\int_1^∞ f(x)dx$ = 5, then which of the following statements must be true about the series $\sum_1^∞f(n)$?</p>
<p>(a) $\sum_1^∞f(n)$ = 0</p>
<p>(b) ... | Math Helper | 374,223 | <p>As the other answers point out quite well, because your function is positive and decreasing the sum will be larger than the integral. To get at your question though: no, this does not defy the integral test. The integral test does not give an upper bound on the sum, but merely says if one (either the sum or integral... |
2,764,073 | <p>I recently was working on a question posted in an AP calculus BC multiple choice sheet which asked:</p>
<p>Let f(x) be a positive, continuous deceasing function. If $\int_1^∞ f(x)dx$ = 5, then which of the following statements must be true about the series $\sum_1^∞f(n)$?</p>
<p>(a) $\sum_1^∞f(n)$ = 0</p>
<p>(b) ... | user2661923 | 464,411 | <p>For visualization purposes, explore $f(x) = (1/x) \Rightarrow \int_{1}^n f(x)dx = ln(n).$<br>
$\sum_{i=1}^{n-1} (1/i) > ln(n) > \sum_{i=2}^n (1/i).$</p>
|
3,091,090 | <p>I came across this question the other day and have been trying to solve it by using some simple algebraic manipulation without really delving into L'Hospital's Rule or the Power Series as I have just started learning limit calculations.
We needed to find :
<span class="math-container">$$\lim_{x \to 0} \frac {x\cos x... | Paramanand Singh | 72,031 | <p>Your approach does not use the laws of limits in proper manner. Each of those laws which deal with limit of sum/difference/product/quotient of two functions has certain restrictions which must be checked before applying them.</p>
<p>In the current question the issue is with application of the quotient rule. You sho... |
563,161 | <p>For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this pretty easily if an inner product space exists. For example, $\|A\| = \sqrt{\langle A,A\rangle}$ and $\|UA\| = \sqrt{\lan... | Elias Costa | 19,266 | <p>By definition of Frobenius norm of a matrix $A\in\mathbb{C}^{n\times n}$,
$$
\| A\|_F:=\sqrt{\mbox{trace}(A^*A)}=\ldots =\sqrt{\sum_{j=i}^n\sum_{i=1}^n\overline{A_{ji}}\cdot A_{ij}}
$$
This norm is a norm defined by an inner product $\langle \cdot , \cdot \rangle$ (<em>i.e.</em> $\|A\|_F:=\sqrt[2]{\langle A , A\ra... |
2,312,096 | <p>How do I compute the integration for $a^2<1$,
$$\int_0^{2\pi} \dfrac{\cos 2x}{1-2a\cos x+a^2}dx=? $$
I think that:
$$\cos2x =\dfrac{e^{i2x}+e^{-2ix}}{2},
\qquad\cos x =\dfrac{e^{ix}+e^{-ix}}{2}$$
But I cannot. Please help me.</p>
| Robert Z | 299,698 | <p>You are on the right track. By letting $z=e^{ix}$ we have that
$$\int_0^{2\pi} \dfrac{\cos 2x}{1-2a\cos x+a^2}\,dx=
\frac{1}{2}\int_{|z|=1}\dfrac{z^2+1/z^2}{1-a(z+1/z)+a^2}\cdot \frac{dz}{iz}\\=\frac{1}{2i}\int_{|z|=1}\dfrac{z^4+1}{z^2(1-za)(z-a)}\,dz
=\pi\left(\mbox{Res}(f,0)+\mbox{Res}(f,a)\right)$$
where we used ... |
346,198 | <p>Recently I was playing around with some numbers and I stumbled across the following formal power series:</p>
<p><span class="math-container">$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$</span></p>
<p>I was able to "simplify" the above expression for <span class="math-container"... | Ivan Meir | 7,113 | <p>Actually the true generalisation for your <span class="math-container">$a=1$</span>, <span class="math-container">$a=2$</span> cases is this expression for your sum: <span class="math-container">$$\frac{1}{a^2}\sum_{s=0}^{a-1}\sum_{r=0}^{a-1}e^{x\omega^r(1+\omega^s)}.$$</span> where <span class="math-container">$\om... |
2,701,128 | <p>I have read multiple definitions so far but something is not clicking.</p>
<p>My most naive understanding is that $|G:H|$ is a "number" (could be infinite) that represents how many times $H$ is in $G$.</p>
<p>But even this doesn't seem fully correct.</p>
<p>I would like a general non-formal explanation and perhap... | Community | -1 | <p>Upgrading my comment to an answer:</p>
<p>You have the right intuition, but I would word it slightly differently.</p>
<p>$|G:H|$ is the number of (left or right) cosets of $H$. A coset is not a copy of $H$; indeed, a coset of $H$ isn't even a subgroup unless it's $H$ itself.</p>
<p>It would be more correct to cal... |
2,917,742 | <p>$\left\{ 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right\} ^ { \frac { 3 } { 2 } } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$</p>
<p>what is the degree and order for above equation </p>
<p>well according to my knowledge the order be should $2$</p>
<p>and degree should be $\frac { 3 } { 2 }$</p>
<p>is my an... | JJacquelin | 108,514 | <p><span class="math-container">$$\left\{ 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right\} ^ { \frac { 3 } { 2 } } = \frac { d ^ { 2 } y } { d x ^ { 2 } } \tag 1$$</span>
The highest order derivative is <span class="math-container">$\frac { d ^ { 2 } y } { d x ^ { 2 } }$</span> . So the order of the ODE is two... |
304,259 | <p>I am stuck on this problem and I'm not sure how to approach it. Can anyone help me out with figuring how to approach the proof?</p>
<p>My task is to:</p>
<blockquote>
<p>Prove that it is impossible to find integers $\,x,\, y\,$ such that $\;2^x = 4y + 3$. </p>
</blockquote>
<p>I assumed a proof by cases would b... | Valtteri | 49,808 | <p>$2^x$ is even and $4 y+3$ is odd...</p>
|
377,471 | <p>If $F[x]$ is a polynomial ring, and $f(x), g(x), h(x)$ and $r(x)$ are four polynomials in it, then is it always true that $f(x)=h(x)g(x)+r(x)$ where $deg(r(x))<deg(g(x))$, or is this true only when $F[x]$ is a Euclidean domain?</p>
<p>Please note that this question has been edited heavily to make it more coheren... | Emmanuel | 85,851 | <p>If F is an Euclidean domain one can always write f(x)=q(x)g(x)+r(x), with r(x)=0 or deg(r(x)) less than deg(g(x)); for it you can read papers about Gröbner basis over Euclidean rings, or simply read the division algorithm in the paper of Franz Pauer name's "Gröbner bases with coefficients in rings".</p>
|
3,051,643 | <blockquote>
<p>Find the volume of intersection of the cylinder<br>
{<span class="math-container">$ x^2 +
y^2 \leq 1 $</span>} , {<span class="math-container">$ x^2 + z^2 \leq 1$</span>}, {<span class="math-container">$ y^2 + z^2 \leq 1$</span>}.</p>
</blockquote>
<p>i am having tough time finding the volume how... | heropup | 118,193 | <p>Perhaps surprisingly, it is not necessary to use polar coordinates, nor even multiple integration or trigonometry, to find the volume.</p>
<p>What we need to do is consider the volume of a solid whose cross-sections perpendicular to the <span class="math-container">$x$</span>-axis are rectangles whose width is <spa... |
11,916 | <p>In <a href="https://mathoverflow.net/questions/11845/theory-mainly-concerned-with-lambda-calculus/11861#11861">Theory mainly concerned with lambda-calculus?</a>, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:</p>
<blockquote>
<p>That would never stick unless there's anoth... | Neel Krishnaswami | 1,610 | <p>So, I'm a computer scientist working in this area, and my sense is the following: </p>
<p>You cannot do good work on functional programming if you are ignorant of the logical connection, period. However, while "proofs-as-programs" is a wonderful slogan, which captures a vitally important fact about the large-scale ... |
1,793,318 | <p>[limit question][1]</p>
<p>Let $x_{k}$ be a sequence of strictly positive real numbers with $\lim \limits_{k \to \infty}\dfrac{x_{k}}{x_{k+ 1}} >1$. Prove that $x_{k}$ is convergent and calculate $\lim \limits_{k \to \infty} x_{k}$.</p>
<p><strong>Attempted answer attached as picture.</strong></p>
<p>I am not... | user341050 | 341,050 | <p>The problem is that $$\lim_{x \rightarrow 0}\; \sum_{n=1}^{\infty} \frac{x}{1 + n^2 x^2}$$ is not $0$, so the function defined by the series isn't continuous. You can see that by evaluating at $x = 1/k:$ $$\sum_{n=1}^{\infty} \frac{1/k}{1 + n^2 / k^2}$$ is a Riemann sum for the integral $$\int_0^{\infty} \frac{1}{1+... |
2,574,768 | <p>There are three planes, <strong>A</strong>, <strong>B</strong>, and <strong>C</strong>, all of which intersect at a single point, <strong>P</strong>. The angles between the planes are given: $$\angle\mathbf{AB}=\alpha$$ $$\angle\mathbf{BC}=\beta$$ $$\angle\mathbf{CA}=\gamma$$ $$0\lt\alpha,\beta,\gamma\le\frac{\pi}{2... | Intelligenti pauca | 255,730 | <p>If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, are unit vectors perpendicular to $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, then $\mathbf{a}\times\mathbf{b}$ is a vector of magnitude $\sin\alpha$ directed along line $\overline{\mathbf{A}\mathbf{B}}$, and so on. The quantity $(\mathbf{a}\times\mathbf{b})\cdot(\mathbf{b}... |
3,941,815 | <p><span class="math-container">$$\sqrt[4]{16} = ?$$</span></p>
<p><span class="math-container">$$16 = |16|(\cos(0)+i\sin(\pi))$$</span>
the equality above holds.</p>
<p><span class="math-container">$$\sqrt[4]{16} = |\sqrt[4]{16}|(\cos(0 + \frac{2k\pi}{4})+i\sin(\pi+\frac{2k\pi}{4})) $$</span>
using De Moivre"s fo... | Deepak | 151,732 | <p>The proper way to express a complex number in polar coordinates is <span class="math-container">$z = |z| (\cos(\mathrm{arg \ z}) + i\sin (\mathrm{arg \ z}))$</span>. Crucially, note that both the cosine and sine are of the same angle, the argument of the complex number.</p>
<p>What you did is wrong because you have ... |
255,483 | <p>How to transform this infinite sum</p>
<p>$$1+\sum_{i\geq1}\frac{x^i}{(1-x)(1-x^2)\cdots(1-x^i)}$$</p>
<p>to an infinite product</p>
<p>$$\prod_{i\geq1}\frac{1}{1-x^i}$$</p>
| Paramanand Singh | 72,031 | <p>The usual technique is to study the function $$f(a, q) = \prod_{n = 1}^{\infty}\frac{1}{1 - aq^{n - 1}}\tag{1}$$ and then express it as a series $$f(a, q) = \sum_{n = 0}^{\infty}x_{n}a^{n}\tag{2}$$ Now we can see that $$f(aq, q) = (1 - a)f(a, q)\tag{3}$$ so that $$\sum_{n = 0}^{\infty}x_{n}a^{n}q^{n} = (1 - a)\sum_{... |
450,410 | <p>I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different.</p>
<hr>
<p><strong>Exercise:</strong> Prove $\lim_{x \to 1} \sqrt{x} = 1$ using $\epsilon$-$\delta$.</p>
<p><strong>My ... | awllower | 6,792 | <p>Your proof is correct.<br>
We can also adopt the following:<br>
Since $|\sqrt x-1|\lt \epsilon$ is equivalent with $1-2\epsilon+\epsilon^2\lt x\lt 1+2\epsilon+\epsilon^2$, we can choose $\delta$ so that $0\lt\delta\lt \min\{|-2\epsilon+\epsilon^2|,|2\epsilon+\epsilon^2|\}$.<br>
Hope this helps. </p>
|
920,050 | <p>The answer is $\frac1{500}$ but I don't understand why that is so. </p>
<p>I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that $x=\frac15$ and plugging in the values I have $\frac1{1-(\frac15)}= \frac54$.</p>
| ReverseFlow | 109,908 | <p>The problem you have is that you do not know why the formula works to begin with. If you did the situation would be clear. Here's the thing:</p>
<p>$$\sum_{n=0}^{\infty}r^n=\frac{1}{1-r} \; \;\;\;\;\; |r|<1.$$</p>
<p>Let $r=\frac{1}{5}$, then you really have the following situation:</p>
<p>$$\left(\frac{1}{5}\... |
29,177 | <p>I had a quick look around here and google before, and didn't find any answer to this particular question, and it's beginning to really irritate me now, so I'm asking here:</p>
<p>How is one supposed to write l (little L), 1 (one) and | (pipe) so that they don't all look the same? One of my teachers draws them all a... | Isaac | 72 | <p>Personally, I always use a script lower-case L and never a print lower-case L in mathematical expressions (my handwriting is such that my lower-case E is sufficiently short and non-script-like that confusing the two is not an issue). As to one versus vertical bar, I typically make my vertical bars extend slightly a... |
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