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 |
|---|---|---|---|---|
1,579,663 | <p>In the first paragraph of <a href="https://en.wikipedia.org/wiki/Finite_field#Definitions.2C_first_examples.2C_and_basic_properties" rel="nofollow">wikipedia:Finite fields</a> they write</p>
<blockquote>
<p>The identity
$$ (x + y)^p = x^p + y^p $$
is true (for every $x$ and $y$) in a field of characteristic ... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\frac{2}{a}\int_{0}^{a}x\sin^{2}\left(\frac{\pi mx}{a}\right)\space\text{d}x=$$
$$a\int_{0}^{a}x\left(1-\cos\left(\frac{2\pi mx}{a}\right)\right)\space\text{d}x=$$
$$a\int_{0}^{a}\left(x-x\cos\left(\frac{2\pi mx}{a}\right)\right)\space\text{d}x=$$
$$a\left(\int_{0}^{a}x\space\text{d}x-\int_{0}^{a}x\c... |
2,879,041 | <blockquote>
<p>Let $f: [a,b] \to \mathbb R$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a) = 0$ and $|f'(x)|\le k|f(x)|$ for some $k$, then $f(x)$ is zero on $[a,b]$. </p>
</blockquote>
<p>I tried proving it using Legrange's Mean Value Theorem but couldn't get it.</p>
<p>$f(x)$ is differentiable ... | Marco Cantarini | 171,547 | <p>Since <span class="math-container">$f\left(a\right)=0$</span> and <span class="math-container">$f $</span> is continuous then exists an interval <span class="math-container">$\left[a,a_{1}\right]$</span> where <span class="math-container">$f$</span> does not change sign. Now assume that <span class="math-container">... |
132,415 | <p>Let $x \in \mathbb{R}^n$</p>
<p>What is</p>
<p>$$\frac{\partial}{\partial x} [ x^Tx ]$$</p>
<p>My guess is: $\frac{\partial}{\partial x} [ x^Tx ] = 0$, because $[x^Tx] \in \mathbb{R}^1$, hence a real number as is interpreted as scalar in this derivation.</p>
| Gavin | 261,086 | <p>Write x as $(x_1, x_2, \cdots, x_n)$. Then $x^t x = \sum_i x_i^2$. So, for example,
$$\frac{d}{dx_1} x^t x = \frac{d}{dx_1} \left( \sum x_i^2\right) = \frac{d}{dx_1} x_1^2 = 2x_1$$
and similarly for each of the other components of $x$. From this it should be clear that
$$\frac{d}{dx} x^t x = 2x^t$$
(The transpose... |
99,018 | <p>If $g$ is Lie algebra over field char(k)=0,
then the following facts are well-known:</p>
<p>1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).</p>
<p>2) S(g)^g and ZU(g)=U(g)^g are isomorphic as commutative algebras. (The <a href="http://en.wikipedia.org/wiki/Duflo_i... | Pasha Zusmanovich | 1,223 | <p>A sort of involved and (seemingly very partial) variant of Duflo isomorphism in characretristic $p$ is claimed in the following paper: N.A. Koreshkov, Central elements and invariants in modular Lie algebras, Russ. Math. (Izv. VUZ) 46 (2002), N7, 20-24 (Russian original is available, for example, at <a href="http://w... |
3,214,255 | <p>How to simplify <span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$</span></p>
<p>Rationalise the denominator </p>
<p><span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$</span></p>
<p>This is still not simplify.</p>
| Hongyi Huang | 619,069 | <p>Hint: <span class="math-container">$6+4\sqrt{2} = (2+\sqrt{2})^2$</span>.</p>
|
3,214,255 | <p>How to simplify <span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$</span></p>
<p>Rationalise the denominator </p>
<p><span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$</span></p>
<p>This is still not simplify.</p>
| nmasanta | 623,924 | <p><span class="math-container">$\sqrt{6+4\sqrt{2}} = \sqrt{(2 + \sqrt{2})^{2}} = {2+\sqrt{2}}$</span> , </p>
<p>as <span class="math-container">$6+4\sqrt{2}= 4 +2 +2.2\sqrt{2} = (\sqrt{2})^{2} + 2 .2. \sqrt{2} + 2^ 2 = (2 + \sqrt{2})^{2}$</span></p>
<p>Now <span class="math-container">$\frac{\sqrt{6+4\sqrt{2}}}{4+2\... |
1,131,323 | <p>I am studying a book on proofs and there are two statements that I don't understand the difference:</p>
<ol>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m + x = m$, then $x = 0$.</p></li>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that... | Tim Raczkowski | 192,581 | <p>In the first statement, $m+x=m$ for all integers $m$. In the second statement, we only need $x+m=m$ for at least one integer $m$.</p>
|
640,680 | <p>Could somebody help me to solve these two unrelated questions?
I have to prove or disprove them. The first one is which I have to answer. The second one is just for me, to understand the topic better.</p>
<p>Prove or disprove the following statements:</p>
<ol>
<li><p>If the sequence $(a_n)_{n\in \mathbb{N}}$ is b... | Module | 114,669 | <p><strong>HINT</strong> For the first one take the sequence $U_{n}=(-1)^{n}$. This sequence clearly is bounded but does it converge? </p>
<p>For the second one: </p>
<p>Let $(U_n)$ be a convergent sequence, and let $\lim (U_n) = u$. Then taking $\epsilon = 1$ we have:</p>
<p>$n > N \implies |U_n - u| < 1$</p... |
1,145,466 | <p>$$y=x^2-5, x∈[-2,0]$$
Here's what I did: $$-2≤x≤0$$
$$x^2≤4 ∧ x^2≤0$$
$$x^2≤0$$
$$x^2-5≤0-5$$
$$y≤-5$$ Is it correct?</p>
| Daniel | 150,142 | <p>If $0\leq a \leq b$, then $0\leq a^2 \leq b^2$. We use that as follows:</p>
<p>Since $-2\leq x \leq 0$, $0 \leq -x \leq 2$ so (applying the first line) $0 \leq (-x)^2 \leq 2^2$, i.e. $0 \leq x^2 \leq 4$. Apply minus five at both sides and you're done.</p>
|
1,145,466 | <p>$$y=x^2-5, x∈[-2,0]$$
Here's what I did: $$-2≤x≤0$$
$$x^2≤4 ∧ x^2≤0$$
$$x^2≤0$$
$$x^2-5≤0-5$$
$$y≤-5$$ Is it correct?</p>
| John | 7,163 | <p>The first and second derivatives of the function are $y' = 2x$ and $y'' = 2$, respectively. From this we can see that the only extremum over the reals is at $x=0$, and that it's a minimum.</p>
<p>Hence, the minimum of the range on the interval $[-2,0]$ is at $x=0$, so $y=-5$ is the minimum. The maximum is at the ... |
1,145,466 | <p>$$y=x^2-5, x∈[-2,0]$$
Here's what I did: $$-2≤x≤0$$
$$x^2≤4 ∧ x^2≤0$$
$$x^2≤0$$
$$x^2-5≤0-5$$
$$y≤-5$$ Is it correct?</p>
| MarnixKlooster ReinstateMonica | 11,994 | <p>To answer this age-old OP-gone question, the following seems the most direct approach.<span class="math-container">$%
\require{begingroup}
\begingroup
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phan... |
3,574,309 | <p>My understanding of a logical formula being valid is that it concludes something, for example:</p>
<p><span class="math-container">$[(A \rightarrow B) \wedge (B \rightarrow C)] \rightarrow [A \rightarrow C]$</span> is logically valid. (I may be mixing this up with rule of inference)</p>
<p>I'm trying to understand... | Graham Kemp | 135,106 | <blockquote>
<p>I am interpreting this 1 as saying that the original formula was logically valid. Is this a correct interpretation? If not, what is this <span class="math-container">$1$</span> telling me? I know that it means that for all possible inputs into the logical expression, the expression holds, does this im... |
8,237 | <p>In this example, I want a series of four buttons to change the value of a variable that is used dynamically to drive a plot. I am trying to figure out why using Table around the buttons causes a problem.</p>
<p>This works:</p>
<pre><code>{Button["1", freq = 1], Button["2", freq = 2], Button["3", freq = 3],
Butt... | Editortoise-Composerpent | 1,451 | <p>Look at the <code>FullForm</code> of the two lists:</p>
<pre><code>{Button["1", freq = 1], Button["2", freq = 2], Button["3", freq = 3], Button["4", freq = 4]} //FullForm
(* List[Button["1",Set[freq,1]],Button["2",Set[freq,2]],Button["3",Set[freq,3]],Button["4",Set[freq,4]]] *)
Table[Button[ToString[i], freq2 = i]... |
3,477,152 | <p>I am trying to prove that</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$</span></p>
<p>diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", but I have no clue about how I would quickly prove this.</p>
<p>So f... | J.G. | 56,861 | <p>Use the <a href="https://en.wikipedia.org/wiki/Term_test" rel="nofollow noreferrer">term test</a>: the <span class="math-container">$n$</span>th term doesn't approach <span class="math-container">$0$</span> as <span class="math-container">$n\to\infty$</span>.</p>
|
2,434,373 | <p><strong>Question</strong></p>
<blockquote>
<p>Using Calculus, find points <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the parabola <span class="math-container">$y=1-x^2$</span> such that an equilateral triangle is formed by the <span class="math-container">$x$</span>-axis a... | Liäm | 189,419 | <p>If I understand correctly, the derivation is straight-forward and you just start with
$$
\nabla \log \pi(s,a) = \frac{\nabla \pi(s,a)}{\pi(s,a)} = \phi(s,a) + \theta
$$
and the $\theta$ should correspond to the expectation. Of course I am using the fact $\pi(s,a) \propto e^{-\phi(s,a)^{T}\theta}$. And the operator $... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| Bernard | 202,857 | <p>This substitution is to be used as a last resort? <em>Bioche's rules</em> say in this case the correct substitution is $u=\sin x$, $\mathrm d\mkern1mu u=\cos x\,\mathrm d\mkern1mu x$. Indeed
$$\int\frac{\mathrm d\mkern1mu x}{\cos x}=\int\frac{\cos x\,\mathrm d\mkern1mu x}{\cos^2 x}=\int\frac{\mathrm d\mkern1mu u}{1-... |
3,332,927 | <p>An analytic function that maps the entire complex plane into the real axis must map the imaginary axis onto:</p>
<p>A) the entire real axis</p>
<p>B) a point</p>
<p>C) a ray</p>
<p>D) an open finite interval</p>
<p>E) the empty set</p>
<p>I was thinking that it might be a constant function.
Any help would be ... | Nitin Uniyal | 246,221 | <p><em>Hint</em>:As a consequence of Open mapping theorem, the dimension of Image space of an analytic <span class="math-container">$f$</span> can be either <span class="math-container">$0$</span> or <span class="math-container">$2$</span>. Moreover if it is <span class="math-container">$0$</span> then <span class="mat... |
115,433 | <p>Mathematica has a lot of machinery for working with predefined probability distributions. It is not clear how to make that machinery work with a new distribution.</p>
<p>Suppose I want to define a brand new distribution</p>
<pre><code>MyDistribution[a, b, c]
</code></pre>
<p>What is the minimum I need to specify ... | wolfies | 898 | <p>Not really an answer ... more like an extended comment that is too long for the comment box. But I found the question interesting, for a number of reasons:</p>
<ol>
<li><p>I did not know that Mma had a <code>FisherInformation</code> function hidden away where you found it - how DID you find it? </p></li>
<li><p>You... |
146,973 | <p>I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent.
I figured out how to do this for the minimum of $n$ variables, but i struggle with doing it for 2 with the maximum.</p>
<p>(The context in whi... | Bravo | 24,451 | <p>The minimum of two independent exponential random variables with parameters $\lambda$ and $\eta$ is also exponential with parameter $\lambda+\eta$. </p>
<p>Also $\mathbb E\big[\min(X_1,X_2)+\max(X_1,X_2)\big]=\mathbb E\big[X_1+X_2\big]=\frac{1}{\lambda}+\frac{1}{\eta}$. Because $\mathbb E\big[\min(X_1,X_2)\big]=\fr... |
4,488,740 | <p>It is known that <span class="math-container">$$\sin^{−1}x+\sin^{−1}y = \sin^{-1}\left[x\sqrt{1 – y^2} + y\sqrt{1 – x^2}\right] $$</span> if <span class="math-container">$x, y ≥ 0$</span> and <span class="math-container">$x^2+y^2 ≤ 1.$</span></p>
<p>I know that the given condition makes sure that <span class="math-c... | Community | -1 | <p>By the first principle definition, we have the derivative,
<span class="math-container">$$\lim_{h\to 0} \dfrac{(x+h)^{2/3} - x^{2/3}}{h}$$</span></p>
<p>I hope you are allowed to use, <span class="math-container">${\lim_{x\to 0}\dfrac{x^{n} - a^{n}}{x-a} = n a^{n-1}} $</span></p>
<p>So,
<span class="math-container">... |
1,442,147 | <p>Could I quickly spot the inverse of a permutation from its 2-cycle composition?</p>
<p>For example, given that $\rho=(1 \ 9)(1 \ 4)(1 \ 5)(1 \ 8)(2 \ 10)(2 \ 3)(2 \ 6)(2 \ 7)$, how to find its inverse from this 2-cycle decomposition?</p>
| Eclipse Sun | 119,490 | <p>Hint: Use $(ab)^{-1}=b^{-1}a^{-1}$ and $(i\, j)=(i\, j)^{-1}$.</p>
<p>Where ab is the composite permutation $(a\circ b)(k)$, $k$ is an element of the set being permuted, and $(i\, j)$ represents the $2$-cycle of the elements $i$ and $j$.</p>
|
938,429 | <p>The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached.</p>
<p><img src="https://i.stack.imgur.com/79dEd.png" alt="This is the question."></p>
<p><img src="https://i.stack.imgur.com/GuZ7z.jpg" alt="This is the solution that I tried."></p>
<p>The problem of my solution i... | E W H Lee | 170,579 | <p>Make the substitution $y = \sqrt{x}$. Then $x=y^2$ so that $dx = 2y\,dy$, and
$$
\int \frac{dx}{x^2+x\sqrt{x}} = \int \frac{2y\,dy}{y^4+y^3} = 2\int \frac{dy}{y^2(y+1)}.
$$
You should be able to carry on from here?</p>
|
2,091,597 | <p>An Opening Note : First of all, I want to make this very clear that by the phrase "without using trigonometry tables", I mean without using them to find $\sin$ values of the "non-standard angles" (For example $73.5^\circ$).</p>
<p>Now, its obviously easy to find a somewhat "broader" range for the answer. Taking the... | Enrico M. | 266,764 | <p>If you want a mental cool approximation, here it is: for $\sin(x)$ you can state that</p>
<p>$$\sin x \approx \frac{x}{60} ~~~~~~~~~~~ x \leq 30^{\circ}$$
$$\sin x \approx \frac{30 + x}{120} ~~~~~~~~~~~ x > 30^{\circ}$$</p>
<p>From this you can derive</p>
<p>$$\cos x = \sin (90-x)$$</p>
<p>and</p>
<p>$$\tan ... |
2,091,597 | <p>An Opening Note : First of all, I want to make this very clear that by the phrase "without using trigonometry tables", I mean without using them to find $\sin$ values of the "non-standard angles" (For example $73.5^\circ$).</p>
<p>Now, its obviously easy to find a somewhat "broader" range for the answer. Taking the... | Claude Leibovici | 82,404 | <p>Suppose that you know the closest angle for which you know the value of trigonometric function. Let us name it $a$. Now develop, around $x=a$, the since function as a Taylor series. You should get</p>
<p>$$\sin(x)=\sin (a)+(x-a) \cos (a)-\frac{1}{2} (x-a)^2 \sin (a)+O\left((x-a)^3\right)$$</p>
<p>Let us apply the... |
2,753,812 | <p>So far I have been looking for a possible divisor such that the sum of $7$ seventh powers will never leave a certain remainder but there seems not to be such a divisor below $400$ and the search is too computationally heavy to continue.</p>
| ViHdzP | 718,671 | <p>The number of ways to make a number less than <span class="math-container">$N$</span> as the sum of seven <span class="math-container">$7$</span>th powers is at most the number of ways to choose seven (not necessarily distinct) integers not exceeding <span class="math-container">$\sqrt[7]{N}$</span>. But <span class... |
3,483,017 | <p>Is there a nice or simple form for a sum of the following form?
<span class="math-container">$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$</span></p>
<p>Motivation: Due to a computation in the formalism of Schubert calculus the above sum with <span class="math-container">$k = \lceil n/2 \rceil -1$</s... | Zhuli | 141,184 | <p>No need for a power tool when a manual tool does the trick.</p>
<p>Hockey-stick identity:</p>
<p><span class="math-container">$$
\sum_{i=0}^k \binom{n+i}{i} = \binom{n+k+1}{k}
$$</span></p>
<p>Applying to our expression:</p>
<p><span class="math-container">$$
1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1... |
2,254,672 | <p>I'm trying to solve a differential equation for the modeling of an ultrasonic horn in wich his form is a catenary, and his differential equation for a wave involves a hyperbolic tangent. I have the solution for the differential equation, but I need to do the "steps by steps". I had transformed the trigonometric coef... | JJacquelin | 108,514 | <p>Appropriate changes of function and variable leads to a second order linear ODE easy to solve :</p>
<p><a href="https://i.stack.imgur.com/bVSC6.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bVSC6.jpg" alt="enter image description here"></a></p>
<p>If $\lambda>1$ change the exponential to si... |
1,131,814 | <p>The question is probably obvious, but is there a sense to say that for exemple $$f:[1,2]\cup[3,4]\longrightarrow \Bbb R$$
is continuous on $[1,2]\cup [3,4]$ or not really ?</p>
| Community | -1 | <p>Yes this makes sense. Recall that a function $f$ is continuous on a set $S\subset \Bbb R$ if $f$ is continuous on every point $x_0\in S$ that's:</p>
<p>$$\forall\epsilon>0\;\exists \delta>0,\quad \forall x\in S:\; |x-x_0|<\delta\implies |f(x)-f(x_0)|<\epsilon$$</p>
|
1,775,166 | <p>If $$T:V\rightarrow V$$ is a bijective linear transformation then can $T^2$ also be a bijective linear transformation? Can $T^n$ also be bijective linear transformation?</p>
| Solumilkyu | 297,490 | <p>In fact, any composite of bijective linear transformations on $V$ is also bijective linear. </p>
<p>For the case that given two bijective linear $T_1,T_2:V\rightarrow V$,
there exist two linear $U_1,U_2:V\rightarrow V$ such that
$$U_1T_1=T_1U_1=U_2T_2=T_2U_2={\rm id}_V.$$
By the associative property of linear tran... |
271,886 | <p>I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$, defined by
$$ \alpha_k := \text{Tr}(\omega^{k})$$
are bi-invariant and define classes in de Rham cohomology.</p>
<... | Eric Schlarmann | 89,992 | <p>I got confused by some of the remarks in the other answers and so decided to work it out for myself. Hopefully, the following is helpful to people who stumble upon this question like me.</p>
<p>One can indeed use transgression as discussed in Jeremy Daniel's answer in order to settle this problem, since the generat... |
314,281 | <p>Let <span class="math-container">$B=-B$</span> be a nowhere dense bounded closed convex set in the Hilbert space <span class="math-container">$\ell_2$</span> such that the linear hull of <span class="math-container">$B$</span> is dense in <span class="math-container">$\ell_2$</span>.</p>
<blockquote>
<p><strong>Q... | Bill Johnson | 2,554 | <p>Your revised assumption is that the norm (rather than semi-norm after the revision) on <span class="math-container">$\ell_2$</span> given by sup-ing against vectors in <span class="math-container">$B$</span> is not equivalent to the usual norm on any finite codimensional subspace. So there is an ON sequence <span c... |
1,328,738 | <p>Let $c$ be a close curve such that $c$ does not intersect itself, $c\in \mathbb{R}^2$ (in the plane), show that for all point $P$ that surrounded in $c$ there are two points $A,B$ on $c$ such that $P$ is in the middle of the interval of $A$ & $B$</p>
<p>As you can see English isn't my first language.</p>
| lhf | 589 | <p>If $\pm a$ are roots of $x^4-2x^3+4x^2+6x-21=0$ then</p>
<p>$a^4-2a^3+4a^2+6a-21=0$</p>
<p>$a^4+2a^3+4a^2-6a-21=0$</p>
<p>Adding these two equations gives you a biquadratic equation for $a$, which you can solve. (Make sure you select the roots of the original equation.)</p>
<p>Once you have found $a$, the other ... |
1,907,758 | <p>In <a href="https://arxiv.org/pdf/1411.5057.pdf" rel="nofollow">this paper</a>, (section 2, page 2)$\mathcal{l}_1$ norm is replaced with reweighted $\mathcal{l}_2$ in an optimization problem. I don't understand how $\lVert x\rVert_1$ is replaced with $x^TWx$ and ow the solution has changed to its weighted version. <... | bartgol | 33,868 | <p>The formula for $x^{k+1}$ comes from constraint optimization, in particular, Lagrange multipliers: given a problem of the form</p>
<p>$$
\min f(x)\\
s.t. g(x)=0
$$
introduce the <em>Lagrangian</em> $\mathcal{L}(x,\lambda)=f(x)+\lambda g(x)$. Then, a solution to the minimization problem is a stationary point for the... |
1,134,978 | <pre><code>Let G be a finite group with no subgroups apart from {1G} and G.
(a) Show that G is cyclic.
(b) Show that the number of elements in G is either 1 or a prime number.
</code></pre>
<p>Any ideas how can I solve the first question?</p>
<p>I was thinking to use Lagrange's theorem stating that the order of G mus... | Alex Wertheim | 73,817 | <p>Suppose $G \neq \{e\}$. Let $g\neq e \in G$, and consider $\langle g \rangle$. Since $\langle g \rangle$ is a subgroup of $G$ and is nontrivial, it must therefore be all of $G$, hence $G$ is cyclic.</p>
<p>Your implication is not quite right. If $|G|$ is prime then $G$ is certainly cyclic, but the converse need not... |
1,134,978 | <pre><code>Let G be a finite group with no subgroups apart from {1G} and G.
(a) Show that G is cyclic.
(b) Show that the number of elements in G is either 1 or a prime number.
</code></pre>
<p>Any ideas how can I solve the first question?</p>
<p>I was thinking to use Lagrange's theorem stating that the order of G mus... | velut luna | 139,981 | <p>If $G$ is trivial, then we are done. If $\exists x\ne1$. Then $\langle x \rangle$ is a nontrivial subgroup, and by the given condition, $\langle x \rangle=G$. So $G$ is cyclic.</p>
|
4,080,680 | <p>If <span class="math-container">$\sum a_n$</span> diverges does <span class="math-container">$\sum \frac{a_n}{\ln n}$</span> necessarily diverge for <span class="math-container">$a_n>0$</span>?</p>
<p>I tried <span class="math-container">$a_n=\frac{\ln n}{n^2}$</span> to try bag an easy counterexample but it turn... | Mike F | 6,608 | <p>If you are willing to chuck any and all monotonicity out the window, then you can prove something much stronger:</p>
<p><strong>Claim:</strong> Given <em>any</em> sequence <span class="math-container">$(b_n)$</span> of positive reals converging to zero, one can find a sequence <span class="math-container">$(a_n)$</s... |
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| ShreevatsaR | 205 | <p>In India "sinh" is pronounced "shine", for reasons I have never known.<br>
"cosh" is pronounced to rhyme with "posh".<br>
"tanh" I don't recall hearing being pronounced; maybe you'd pronounce "tanh x" as "shine x by cosh x". :-)</p>
|
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| GEdgar | 442 | <p>In your lecture, pronounce it "hyperbolic sine" the first time, then after that use whatever short form you like.</p>
|
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| gabble1 | 124,295 | <p>My school pronounces them as 'shine' 'cosh' and 'than'. Where in 'than', the 'th' is pronounced as the 'th' in 'thyme', so a soft 'th' sound.</p>
|
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | Henry | 6,460 | <p>I would have thought that bright students, who knew <span class="math-container">$1+x+x^2+\cdots +x^{n-1}= \frac{x^n-1}{x-1}$</span> as a geometric series formula, could say </p>
<p><span class="math-container">$$\dfrac{P(x)}{Q(x)} =\dfrac{x^{104}+x^{93}+x^{82}+x^{71}+1}{x^4+x^3+x^2+x+1}$$</span></p>
<p><span cla... |
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | Hari Shankar | 351,559 | <p>This may be accessible to a high school student:</p>
<p><span class="math-container">$x^{104}+x^{93}+x^{82}+x^{71}+1$</span></p>
<p><span class="math-container">$ = (x^{104}-x^4)+(x^{93}-x^3)+(x^{82}-x^2)+(x^{71}-x)+(x^4+x^3+x^2+x+1)$</span></p>
<p><span class="math-container">$=x^4(x^{100}-1)+x^3(x^{90}-1)+x^2(x... |
785,441 | <p>Prepping for a comprehensive test in August and I am working on a problem from Royden 4th ed. (Chapter 2, #15): Show that if $E$ has finite measure and $\varepsilon>0$, then $E$ is the disjoint union of a finite number of measurable sets, each of which has measure at most $\varepsilon$.</p>
<p>Here is what I hav... | xbh | 514,490 | <p>I have another solution, which seems [at least] easier for me. Of course the knowledge I used exceeds the scope of that specific section. Any comments and criticism are welcome. </p>
<p>Consider a function
<span class="math-container">$$
\newcommand{\abs}[1]{\left\vert #1 \right\vert}
\newcommand\rme{\mathrm e}
\ne... |
990,418 | <p>What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. </p>
<p>I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also <a href="http://www.wolframalpha.com/input/?i=max+sin(a)%2Bsin(b)%2Bsin(c)... | Mark Bennet | 2,906 | <p>Here is a hint, which should get you most of the way there</p>
<p>Note that $\sin B+\sin C= 2\sin \frac {B+C}2 \cos \frac {B-C}2$</p>
<p>If $A$ is fixed then $B+C$ is fixed, and the product is greatest when $B=C$</p>
|
101,566 | <p>How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works?</p>
| Marc van Leeuwen | 18,880 | <p>Assuming you mean to solve $\phi(n)=12$, you can use the formula for $\phi(n)$ that consists of factoring $n$ and then replacing one copy of every prime $p$ that occurs by $p-1$. You need such $p-1$ to divide $12$, so the only $p$ that can occur are $2,3,5,7,13$, and since occurrence with multiplicity at least $2$ a... |
2,434,812 | <p>I am only concerned with the integral of a measurable function $f : M \to \mathbb{R}^{\ast}_{+}$, where $\mathbb{R}^{\ast}_{+}$ contains all non-negative real numbers and positive infinity. As I understand it, this is defined as
$$\sup \left\{ \int s : \text{$s$ is simple and $s\le f$}\right\}.$$</p>
<p>My question... | drhab | 75,923 | <p>No.</p>
<p>Let $f:(0,1]\to[0,+\infty]$ be prescribed by $x\mapsto x^{-\frac12}$ </p>
<p>Then $\int fd\lambda=\left[2x^{\frac12}\right]^1_0=2$ and this is the outcome of the supremum in your question.</p>
<p>However if $s$ is a simple function then its image is finite so that we can find a value $y>0$ such that... |
2,940,072 | <p><span class="math-container">$$\left(\frac{f}{g}\right)'(x_{0})=\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g^2(x_0)}$$</span></p>
<p>So, <span class="math-container">$\frac{1}{g}.f=\frac{f}{g}$</span>, then <span class="math-container">$$\frac{f}{g}'(x_0)=\frac{f(x)\frac{1}{g(x)}-f(x_0)\frac{1}{g(x_0)}}{x-x_0}=f(x)\frac{\f... | Claude Leibovici | 82,404 | <p>There is another way to do it : logarithmic differentiation
<span class="math-container">$$y=\frac{f(x)}{g(x)}\implies \log(y)=\log(f(x))-\log(g(x))$$</span> Differentiate both sides
<span class="math-container">$$\frac{y'}y=\frac{f'(x)}{f(x)}-\frac{g'(x)}{g(x)}=\frac{f'(x)\,g(x)-f(x)\,g'(x) } {f(x)\,g(x) }$$</span>... |
2,887,858 | <p>I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. This question is related to <a href="https://math.stackexchange.com/questions/2887371/calculating-the-surface-integral-int-s-10y-j-d-sigmay">Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$</a> where I try to comp... | Batominovski | 72,152 | <p>MathJax is killing my slow browser, so I have to write a separate answer. All notations can be found in my first answer.</p>
<hr>
<p><strong>Solution Following the OP's Idea</strong></p>
<p>Without loss of generality, suppose that $j=n$. First, it is easy to prove that
$$\text{d}\sigma_{n-1}(\mathbf{y})=\frac{... |
1,046,961 | <p>Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ <br/>
I'm thinking; <br/>
Let $f(x)=-f(2x)$ <br/>
Use a substitution $x=y/2$ for $y \in \mathbb{R}$. <br/>
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$ <br/>
Im just not sure if this is a good approac... | mfl | 148,513 | <p>Note that $$f(0)+f(2\cdot 0)=f(0)+f(0)=0\implies f(0)=0.$$</p>
<p>Consider $x_0\ne 0.$ It is</p>
<p>$$f(x)+f(2x)=0\underbrace{\implies}_{x=x_0/2} f(x_0/2)=-f(x_0),$$</p>
<p>$$f(x)+f(2x)=0\underbrace{\implies}_{x=x_0/4} f(x_0/4)=-f(x_0/2)=f(x_0).$$</p>
<p>By induction, $f(x_0/2^n)=-f(x_0)$ if $n$ is odd and $f(x_... |
4,240,794 | <p>I was trying to obtain the square root of a matrix through the eigenvalues and eigenvectors, and there is something that doesn't add up in some of the demonstrations that I observed after getting stuck.</p>
<p>So, being <span class="math-container">$Q$</span> the eigenvector column matrix of <span class="math-contai... | Barry Cipra | 86,747 | <p>It's convenient to consider the core expression <span class="math-container">$-2x\exp(-2x)-\exp(-2x)+1$</span>, let <span class="math-container">$u=-2x$</span>, and rewrite this function as</p>
<p><span class="math-container">$$f(u)=(u-1)e^u+1$$</span></p>
<p>We see that <span class="math-container">$f(0)=0$</span>,... |
1,842,537 | <blockquote>
<p>In <span class="math-container">$\triangle ABC$</span>, prove:
<span class="math-container">$$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$</span>
for circumradius <span class="math-container">$R$</span>, inradius <span class="math-container">$r$</span>, and exr... | Jack D'Aurizio | 44,121 | <p>We will use the <span class="math-container">$\sum_{cyc}$</span> notation for cyclic sums:
<span class="math-container">$$
\sum_{cyc}f(a,b,c) = f(a,b,c)+f(b,c,a)+f(c,a,b)
$$</span>
for any function <span class="math-container">$f$</span>. Also, let <span class="math-container">$\Delta$</span> be the area of triangle... |
1,842,537 | <blockquote>
<p>In <span class="math-container">$\triangle ABC$</span>, prove:
<span class="math-container">$$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$</span>
for circumradius <span class="math-container">$R$</span>, inradius <span class="math-container">$r$</span>, and exr... | lab bhattacharjee | 33,337 | <p>Using <a href="https://en.wikipedia.org/wiki/Law_of_sines" rel="nofollow noreferrer">Sine Rule</a></p>
<p>and
<span class="math-container">$r_a=4R\sin\dfrac A2\cos\dfrac B2\cos\dfrac C2,$</span>
<span class="math-container">$r=4R\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2,$</span> </p>
<p><span class="math-container">... |
1,532,401 | <blockquote>
<p>If $X\sim \exp(\lambda)$, then $E[X\mid X>20]=20+E[X]$</p>
</blockquote>
<p>Can anyone give an intuitive proof of this property? In the context that $X$ is exponential and thereby memoryless. I am having trouble in visualizing continuous memoryless distributions. Any intuitive example for the sam... | Graham Kemp | 135,106 | <p>An exponential distributed random variable is the measure of waiting time until the arrival of some event, the likes of which occur <em>independently</em> and at some <em>constant average rate</em> (ie: a Poisson Process).</p>
<p>This means that at every point in time the chance of such an event happening is the sa... |
1,984,190 | <p>$X \in \mathbb{R}$ is the number of coin tosses it takes to get the first head.</p>
<p><strong>Question</strong>: find the probability of getting the first head at an even number of tosses.</p>
<p><strong>Official solution</strong>:
$P(X_{even}) = \frac{1}{2^{2}} + \frac{1}{2^{4}} + ... = \frac{1}{3}$ (sum of a ge... | Cato | 357,838 | <p>there is 0.5 chance that you will immediately get a head on toss 1 - but that is not the only way you can fail, you can also get a head on 3,5,7... and fail</p>
<p>after two opening tails (probability 1/4), the game is back where it started - the probability of winning is prob of getting head on toss 2 (if there is... |
2,379,881 | <p>I want to know if this integral converges or not : $\int _{1}^{\infty }\!{\frac {\sin \left( \cos \left( x \right) +\sin\left( x\sqrt {3} \right) \right) }{x}}{dx}.$ I tried to integrate by parts or to use dirichlet's test, but it seems impossible to prove that $\int _{1}^{x}\!\sin \left( \cos \left( t \right) +\si... | Community | -1 | <p>The function $\varphi(t) = \cos(t)+\sin(t\sqrt{3})$ is uniformly almost periodic and has mean value $0$. It's easy to see that $\sin\varphi(t)$ has the same properties. Unfortunately, that doesn't necessarily mean that $\int^x_1\sin\varphi(t)\,dt$ is bounded. If it is, it's a uniformly almost periodic function of $x... |
2,907,378 | <p>In my Math book I'm solving a case where this is the situation:</p>
<p>"The demand curve for good X is linear. At a price (p) of 300 the demand is 600 units. At a price of 680 the demand is 220 units. Also the supply curve for good X is linear. If the price is 400 then the supply equals 200 units, whereas for a pri... | gandalf61 | 424,513 | <p>You have your $x$ and $y$ co-ordinates the wrong way round. Supply is your $y$ co-ordinate (dependent variable) and Price is you $x$ co-ordinate (independent variable) so you should have</p>
<p>$\Delta y = 1000 - 200 = 800$</p>
<p>$\Delta x = 800 - 400 = 400$</p>
<p>$\text{Slope } = \frac{\Delta y}{\Delta x} = \f... |
89,987 | <p>In the sequence: </p>
<p>$$\lim _{n\rightarrow \infty }{\frac {n+1}{2\,n+3}}\neq 3$$</p>
<p>I know how to prove that the limit is actually $1/2$, but is there another way to prove that 1 is NOT the limit? </p>
<p>I tried to prove by negation showing that if 1 is the limit I can't find an $N$ that for every epsilo... | Zev Chonoles | 264 | <p>Let $X$ be some number <em>other</em> than $\frac{1}{2}$. Let's define $d=|X-\frac{1}{2}|$, and because $X\neq\frac{1}{2}$ we have $d>0$. </p>
<p>The <a href="http://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality" rel="nofollow">reverse triangle inequality</a> says that for any $a$ and $b$... |
4,610,313 | <p>I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:</p>
<blockquote>
<p>Given that <span class="math-container">$\log_{4n} 40\sqrt{3} = \log_{3n} 45$</span>, find <span class="math-container">$n^3$</span> (MA<span class="math-container">$\Theta$</span> 1991).</p>
</blockquot... | albert chan | 696,342 | <p><span class="math-container">$\displaystyle\frac{\ln(40\sqrt{3})}{\ln(4n)} = \frac{\ln(45)}{\ln(3n)}$</span></p>
<p><span class="math-container">$\displaystyle \frac{\ln(3n)}{\ln(4n)}
= \frac{\ln(45)}{\ln(40\sqrt{3})} × \frac{k}{k}
= \frac{\ln(45^k)}{\ln[(40\sqrt{3})^k]}$</span></p>
<p>Find k, such that ratio inside... |
254,236 | <p>If a matrix $A$ is diagonalizable, is $A$ invertible? </p>
<p>I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.</p>
| Jonathan | 37,832 | <p>If that diagonal matrix has any zeroes on the diagonal, then $A$ is not invertible. Otherwise, $A$ is invertible. The determinant of the diagonal matrix is simply the product of the diagonal elements, but it's also equal to the determinant of $A$.</p>
|
2,618,813 | <p>What is the image of the set {$ { {z ∈ C : z = x + iy, x ≥ 0, y ≥ 0} } $} under
the mapping $ z \to z^2$</p>
<p>my answer : $f(z) = z^2 =(x+iy)^2 = x^2-y^2 +2ixy$,</p>
<p>here I get $u=x^2-y^2$ and $v=2xy$. After that I am confused that how can I find the image of the set.</p>
<p>Please help me, as any Hints can... | Peter Szilas | 408,605 | <p>For fun:</p>
<p>Domain : </p>
<p>$D= ${$z| z=x+iy,$; $x\ge 0$, $ y\ge 0$}.</p>
<p>First quadrant in the complex plane.</p>
<p>Let $z= re^{i\theta}$, with </p>
<p>$r \in \mathbb{R^+}$; $0 \le \theta \le π/2.$</p>
<p>$f(z) := z^2= r^2e^{i2\theta}$, </p>
<p>$r \in \mathbb{R^+}$; $0\le 2\theta \le π.$</p>
<p>Im... |
1,481,627 | <p>I have been given homework in which the $2\times 2$ matrices of determinant $1$ are equipped with the subspace topology of $\mathbb{R}^4$. However, $\mathbb{R}^4$ is a space of 4-tuples, while 2x2 matrices are not n-tuples. How do I get the open sets of this topology?</p>
<p>How is this set of matrices a subset of ... | Joel Cohen | 10,553 | <p>The space of $2 \times 2$ matrices is identified with $\mathbb{R}^{2^2}$ via</p>
<p>$$\left(\begin{matrix}a&b\\c&d\end{matrix}\right) \longmapsto \left(\begin{matrix}a\\b\\c\\d\end{matrix}\right)$$</p>
<p>We deduce the topology from this identification. So your subset of matrices is identified with $\{(a,b... |
268,308 | <p>In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic? </p>
| Asaf Karagila | 622 | <p>It depends on the context. My masters thesis, for example, began with saying that I expect the reader to be familiar with forcing. I am not expecting the reader to be familiar with other topics which are relatively common, though. These topics are fully explained in my thesis.</p>
<p>When writing something one can ... |
357,847 | <p>Let <span class="math-container">$n\ge 3$</span> be an integer. I would like to know if the following property <span class="math-container">$(P_n)$</span> holds: for all real numbers <span class="math-container">$a_i$</span> such that <span class="math-container">$\sum\limits_{i=1}^na_i\geq0 $</span> and <span clas... | Per Alexandersson | 1,056 | <p>This is just a long comment, but translating to the notation of symmetric functions, you ask if
whenever <span class="math-container">$e_{111}(x) \geq 0$</span> and <span class="math-container">$e_3(x) \geq 0$</span>, we have
<span class="math-container">$$
n^2 p_{(3)}(x) \geq p_{111}(x).
$$</span>
This latter is eq... |
3,636,563 | <p>How to find the values of <span class="math-container">$x,y$</span> and <span class="math-container">$z$</span> if
<span class="math-container">$3x²-3(1126)x=96y²+24(124)y=8z²-4(734)z $</span>?</p>
<p>I dont have any idea!! I think we can have many values of <span class="math-container">$x,y$</span> and <span clas... | Community | -1 | <p><span class="math-container">$$\sum_{k=2}^{\infty}\left(\frac{1}{k-1}\:-\:\frac{1}{k}\right) = 1$$</span></p>
<p>This sum is simply equal to <span class="math-container">$1$</span>.</p>
|
406,966 | <p>Define a graph with vertex set <span class="math-container">$\mathbb{R}^2$</span> and connect two vertices if they are unit distance apart. The famous <a href="http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem" rel="nofollow noreferrer">Hadwiger-Nelson problem</a> is to determine the <a href="http://en.wi... | Asaf Karagila | 622 | <blockquote>
<p>Is $\pi$ a normal number?</p>
</blockquote>
<p>All we know is that rational numbers are not normal, and that $\pi$ is irrational.</p>
<p>You can just as well replace $\pi$ by many other constants, and we don't know much about any of them.</p>
|
406,966 | <p>Define a graph with vertex set <span class="math-container">$\mathbb{R}^2$</span> and connect two vertices if they are unit distance apart. The famous <a href="http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem" rel="nofollow noreferrer">Hadwiger-Nelson problem</a> is to determine the <a href="http://en.wi... | user642796 | 8,348 | <p>Here's a little ditty from Complexity Theory.</p>
<p>The following facts are quite easy to prove:</p>
<ol>
<li>$\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P} \subseteq \mathsf{NP} \subseteq \mathsf{PSPACE} \subseteq \mathsf{EXPTIME} \subseteq \mathsf{EXPSPACE}$;</li>
<li>$\mathsf{L} \neq \mathsf{PSPACE}$;<... |
1,861,292 | <p>I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a subgroup of $R\otimes B$, but he doesn't give an example, just states "the free abelian groups and equivalence relations... | Alex Provost | 59,556 | <p>The problem is that the "obvious" inclusion mapping $I \otimes B \to R \otimes B$, given by tensoring the inclusion mapping with the identity, need not be injective. Recall that, by definition, a module is <em>flat</em> if tensoring with that module preserves the head of exact sequences; but that is the same as sayi... |
2,328,758 | <p>The Equation
$11^x + 13^x + 17^x =19^x $</p>
<p>Has </p>
<ol>
<li>No Real Roots</li>
<li>Only One Real Roots</li>
<li>Exactly Two Real Roots </li>
<li>More than Two Real Roots </li>
</ol>
<p>What I have done is </p>
<p>The function $f(x)=11^x + 13^x + 17^x -19^x $ is strictly increasing and being always positi... | Arnaldo | 391,612 | <p>I'm afraid your answer is wrong. </p>
<p>One way to see that is consider the function</p>
<p>$$f(x)=\left(\frac{11}{19}\right)^x+\left(\frac{13}{19}\right)^x+\left(\frac{17}{19}\right)^x-1$$</p>
<p>is strictly decreasing and $f(-\infty)=\infty$, $f(\infty)=-1$.</p>
<p>So there is only ONE real root.</p>
|
2,852,248 | <p><em>There are 5 classes with 30 students each. How many ways can a committee of 10 students be formed if each class has to have at least one student on the committee?</em></p>
<p>I figured that we first have to choose 5 people from each class, so there are $10^5$ options. There remain total of 29*5=145 students to ... | Boyku | 567,523 | <p>and here is via weighted species and e.g.f.</p>
<p>Let <span class="math-container">$X, Y, Z, U, V$</span> be five sorts of students</p>
<p>Then <span class="math-container">$ A = X \cdot E_{29}(X) + E_2(X) \cdot E_{28}(X) + E_3(X) \cdot E_{27}(X) + E_4(X) \cdot E_{26}(X) + E_5(X) \cdot E_{25}(X) + E_6(X) \cdot E_... |
983,923 | <p>Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$,
$$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective.</p>
<p>But why for the inverse, suppose that $E$ and $F$ is the subset of $D$,
$$f^{-1}(E \cap F) = f^{-1}(E) \cap f^{-1}(F)$$
without saying that the inverse functi... | Marc van Leeuwen | 18,880 | <p>The sentence "This means there exists elements $y_1\in E$ and $y_2\in F$" is incomplete. The mere fact that $y_1,y_2$ exist is not very useful. Say what property those elements have, and you will see why they are equal.</p>
|
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| Narasimham | 95,860 | <p>Just like when you use an eraser on pencil written paper stuff, erase $b$ and in its place write $B+C$ wherever it comes.</p>
<p>You know
$$\tan(A+b)=\frac{\tan A+\tan b}{1-\tan A\tan b}$$</p>
<p>It becomes:</p>
<p>$$\tan(A+B+C)=\frac{\tan A+\tan (B+C)}{1-\tan A\tan (B+C)}$$</p>
<p>Now expand and simplify... tha... |
2,900,112 | <p>Let $f:\mathbb{R}^n \to \mathbb{R}$. Let $x_0 \in \mathbb{R}^n$. Assume $n-1$ partials exist in some open ball containing $x_0$ and are continuous at $x_0$, and the remaining $1$ partial is assumed only to exist at $x_0$. A well known result states that this implies $f$ is differentiable at $x_0$. </p>
<p>My questi... | David C. Ullrich | 248,223 | <p>It's been said that (i) in $\Bbb R^2$ the mere existence of both partials is not enough, (ii) it follows that continuity of $n-2$ partials in $\Bbb R^n$ is not enough.</p>
<p>This is about (i); for the benefit of any readers to whom it's not obvious, it really <em>is</em> obvious.</p>
<p>The point being that by de... |
2,586,466 | <p>Write the value of $n$ if the sum of n terms of the series $1+3+5+7...n =n^2$.</p>
<p>I'm not getting the right value if I proceed with the general formula for finding sum of n terms of a arithmetic series. The general summation formula for arithmetic series is $\frac{n(2a+(n-1)d)}{2}$, where $a$ is the first term,... | ArsenBerk | 505,611 | <p>If we add $2+4+...+(n-1) = 2(1+2+3+...+\frac{n-1}{2})$ to both sides of the equation, we get
$$1+2+3+...+n = n^2+2\bigg(1+2+3+...+\frac{n-1}{2}\bigg)$$
which can be written as:
$$\frac{n(n+1)}{2} = n^2+2\bigg[\frac{\big(\frac{n-1}{2}\big)\big(\frac{n+1}{2}\big)}{2} \bigg]$$
that is
$$\frac{n^2+n}{2} = n^2+\frac{n^2... |
2,738,679 | <p>I am having trouble to prove this exercise for a Real Analysis course. I tried two approaches. However, I am insecure about both of them.</p>
<p>This is the question:</p>
<p>If $X$ is a partition of ℕ* such that:</p>
<p>i) $1 ∈ X$;</p>
<p>ii) For all $n ∈ ℕ$*, if $n ∈ X$, then $2n ∈ X$;</p>
<p>iii) For all $n ∈... | mathcounterexamples.net | 187,663 | <p>First, I assume that when you say <em>$X$ is a partition of $\mathbb N^*$</em>, you mean that $X$ is a subset of $\mathbb N^*$.</p>
<p><strong>"Obviously" $X \subseteq \mathbb N^*$. If would be good that you provide simple arguments on the why.</strong></p>
<p>Let's prove conversely that $\mathbb N^* \subseteq X$... |
2,293,838 | <p>I have question(1) in the following picture:</p>
<p><a href="https://i.stack.imgur.com/ZwSLl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZwSLl.png" alt="enter image description here"></a></p>
<p>and the solution to it is given in the following picture:</p>
<p><a href="https://i.stack.imgur.... | N. Shales | 259,568 | <p>The palindromic words are either</p>
<ul>
<li>odd length with either h, c or a as the central letter.</li>
</ul>
<p>or</p>
<ul>
<li>even length with no central letter.</li>
</ul>
<p>In both cases we can <em>build</em> a palindrome by successively putting pairs of h's, c's and a's either side of a smaller palindr... |
105,583 | <p>this is a simple question, and excuse me if it's already been answered; I searched around and couldn't find anything.</p>
<p>I have two listplots, both along the same number of x data points, but with different y values. I want to find the difference between the two y values, while keeping the x values the same. I ... | Brett Champion | 69 | <p>Turn the data into time series, and do the arithmetic with them:</p>
<pre><code>ts1 = TimeSeries[{{1, 2}, {2, 3}, {3, 5}, {4, 7}, {5, 11},
{6, 13}, {7, 17}}];
ts2 = TimeSeries[{{1, 3.87}, {2, 3.53}, {3, 3.40}, {4, 3.33},
{5, 3.25}, {6, 4.25}, {7, 5.24}}];
Normal[ts1 - ts2]
</c... |
181,327 | <p>If $q$ and $w$ are the roots of the equation $$2x^2-px+7=0$$ </p>
<p>Then $q/w$ is a root of ? </p>
<p>P.s:- It is an another question of <a href="https://math.stackexchange.com/questions/181305/how-do-i-transform-the-equation-based-on-this-condition">How do I transform the equation based on this condition?</a></p... | mirror2image | 25,050 | <p>if $B > 0$ by dividing on B you can bring <em>all</em> your constraint, both equality and inequalities, to single "unit simplex constraint" There are a lot of papers for optimization on unit simplex. Some of them are using solving unconstrained problem and projecting it projection on unit simplex each iteration ... |
1,015,137 | <p>I have an exercise book from my university which doesn't specify a quantifier.</p>
<p>It uses expressions like "here $A$,$B$,$C$ are sets", or "if $x \notin A$ then ..." (it uses $x$ before it is even defined, just out of nowhere).</p>
<p>I'm going to need an answer from my university, of course, but I want to ask... | GEdgar | 442 | <p>How about something more elementary?</p>
<p>If $x_i=0$ for all $i$, it is easy. So assume not.</p>
<p>Let $M = \max\{|y_1|,|y_2|,\dots\}$. If $M=\infty$ it is easy, so assume not.</p>
<p>Certainly $\sum_{i=1}^\infty |y_i| \ge M$.</p>
<p>For all $i$, we have $|x_i y_i| \le |x_i|\; M$, so
$$
\sum_{i=1}^\infty |x... |
23,639 | <p>Prove that for every positive integer $x$ of exactly four digits, if the sum of digits is divisible by $3$, then $x$ itself is divisible by 3 (i.e., consider $x = 6132$, the sum of digits of $x$ is $6+1+3+2 = 12$, which is divisible by 3, so $x$ is also divisible by $3$.)</p>
<p>How could I approach this proof? I'm... | Bill Dubuque | 242 | <p>It's due to radix representation being <em>polynomial form</em> in the radix, e.g. $\rm\ n = 4321 = p(10)\ $ for $\rm\ p(x) = 4\: x^3 + 3\: x^2 + 2\: x + 1\:.\:$ Thus mod $\rm\:3\::\ 10\equiv 1\ \Rightarrow\ p(10)\equiv p(1)\ =\ \sigma(n) :=$ sum of digits. Aternatively one may simply put $\rm\ x = 10\ $ in the $\:$... |
1,645,865 | <p>I am ultimately trying to prove, for an Exercise in Burton's Elementary Number Theory, that $x^4 - y^4 = 2z^2$ has no solution in the positive integers.</p>
<p>I can establish that if there is a solution, the solution with the smallest value of x has gcd(x,y)=1 </p>
<p>I see that $x^4 - y^4 = (x^2 + y^2)(x^2 - y^2... | AnalysisStudent0414 | 97,327 | <p>Call $f(x)=x^2 e^x -1$, and note that it is continuous. Since
$$\frac{d}{dx} (x^2 e^x -1) = e^x(x^2+2x)$$</p>
<p>we have that $(0,-1)$ is a maximum and $(-2,4e^{-2}-1)$ is a minimum.</p>
<p>Since $4e^{-2}-1<0$ and $\displaystyle \lim_{x\to -\infty} f(x)=-1$ we know that $f$ is negative in $(-\infty, -2)$. We a... |
1,645,865 | <p>I am ultimately trying to prove, for an Exercise in Burton's Elementary Number Theory, that $x^4 - y^4 = 2z^2$ has no solution in the positive integers.</p>
<p>I can establish that if there is a solution, the solution with the smallest value of x has gcd(x,y)=1 </p>
<p>I see that $x^4 - y^4 = (x^2 + y^2)(x^2 - y^2... | Community | -1 | <p>Since the statement is always positive $x^2$ is positive $e^x$ is positive we can take the logarithm and have more tangible:</p>
<p>$$x+2\log(|x|)=0$$</p>
<p>Now we have $\lim\limits_{x \to 0} x+2\log(|x|)=-\infty$. Observe that this is so for both sides of $0$: $0^{+}$ and $0^{-}$. So we are interested what is ha... |
80,432 | <p>I know Mathematica's if format is </p>
<pre><code>If[test, then result, else alternative]
</code></pre>
<p>For example, this</p>
<pre><code>y:=If[RandomReal[]<0.2, 1, 3.14]
</code></pre>
<p>would take a random real number between $0$ and $1$, and evaluate it. If it's less than $0.2$, it'll map <code>y</code> ... | ciao | 11,467 | <p>I'd use <code>Interval</code> for something like this.</p>
<p>E.G., a function that takes as arguments the intervals, what a "hit" in the interval should return, and the target:</p>
<pre><code>f1 = Pick[#2, IntervalMemberQ[Interval /@ #1, #3]] &;
f1[{{0, .1}, {.1, .2}, {.2, .3}}, {1, 2, 3}, .23]
(* {3} *)
</... |
3,490,198 | <p>For all <span class="math-container">$a,b \in \mathbb{Z}$</span> and for all <span class="math-container">$m,n \in \mathbb{N}\setminus \left\lbrace0\right\rbrace$</span>,</p>
<p>is <span class="math-container">$a^{48m+1}+b^{48n+1} \equiv 0 \pmod{39} \iff a+b \equiv 0 \pmod{39}$</span>?</p>
<p>I think the answer is... | J. W. Tanner | 615,567 | <p>Note that <span class="math-container">$39=3\times13$</span>, and <span class="math-container">$3$</span> and <span class="math-container">$13 $</span> are prime.</p>
<p>Using Fermat's little theorem, <span class="math-container">$a^3\equiv a\bmod 3$</span>, </p>
<p>and by induction <span class="math-container">$a... |
987,997 | <p>A set of formulas is independent if no proper subset is logically equivalent to it.</p>
<p>Note that this exercise appears in Enderton 1.2 10(c) and is marked as star.</p>
| Mark Fischler | 150,362 | <p>The proof for any <em>finite</em> set of formulas is easy enough:</p>
<p>Assume set of formulas $S$ is not itself an independent set (if it is, then clearly $S$ meets the condition of the theorem since $S$ is obviously equivalent to itself). Let the number of formulas in $S$ be $n_0$.</p>
<p>Since $S$ is not an in... |
987,997 | <p>A set of formulas is independent if no proper subset is logically equivalent to it.</p>
<p>Note that this exercise appears in Enderton 1.2 10(c) and is marked as star.</p>
| hmakholm left over Monica | 14,366 | <p>For an infinite <span class="math-container">$S$</span> we cannot stick to getting an equivalent and independent <em>subset</em> of <span class="math-container">$S$</span>. An example that shows this is
<span class="math-container">$$ \begin{align} S =\{ & A_1 ,\\
& A_1 \land A_2 ,\\
& A_1 \land A_2 \l... |
1,166,382 | <blockquote>
<p>Evaluate the integral $\int_0^1 \cos(\ln(x)) \, dx$</p>
</blockquote>
<p>I was able to evaluate the improper integral which is:</p>
<p>$$\frac{x\left(\sin \ln x + \cos \ln x\right)}{2}$$</p>
<p>I was using the substitution $u = \ln x$, and afterward I did integration by parts twice and got the resu... | Matthew Cassell | 181,563 | <p>Set</p>
<p>$$\begin{align}
u &= \ln(x) \implies e^{u}du = dx \\
x &= 0 \implies u = -\infty \\
x &= 1 \implies u = 0
\end{align}$$</p>
<p>Hence, you get</p>
<p>$$I = \int_{-\infty}^{0} e^{u}\cos(u) du$$</p>
<p>Integrating by parts twice, first with $v = e^{u}$, $w' = \cos(u)$ and secondly with $v = e... |
243,849 | <p>I was working on an examples from my textbook concerning transforming formulae into disjunctive-normal form (DNF) until I found an expression that I cannot solve. I hope somebody can help me transform the following statement into DNF:</p>
<p>$$ (\lnot q \lor r) \land ( q \lor \lnot r)$$</p>
| William Macrae | 49,817 | <p>Think of it as an expression with $+$ and $\cdot$ and use the distributive law (I'll use $q'$ as notation for $\neg q$:</p>
<p>$(q' + r)(q + r') = q'q + q'r' + rq + rr' = q'r' + rq$</p>
<p>I cancelled $q'q$ and $r'r$ because $q$ cannot be true and false (and same for $r$). With your symbols, this gives</p>
<p>$(\... |
760,739 | <p>Find the limit of the sequence given by
$$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$</p>
<p>I think the answer is $\frac{20}{7}$ after dividing, but is that right? </p>
| Mark Bennet | 2,906 | <p>Which limit do you want? As $n \to 0$ this goes to the limit $-\frac 12$.</p>
<p>I put this as an answer because the other answers were up, and there are alternatives to those.</p>
|
3,443,711 | <p>I'm not sure I get this.</p>
<p>Because the sine function is an odd function, for
a negative number u, sin2u= -2sinucosu</p>
<p>Is it true or false and why?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>You can use <span class="math-container">$$a=\frac{b\sin(2\beta)}{\sin(\beta)}=2b\sin(\beta)$$</span> and <span class="math-container">$$c=\frac{\sin(\pi-2\beta)}{\sin(\beta)}$$</span> so you will get
<span class="math-container">$$p=b\left(2\sin(\beta)+1+\frac{\sin(\pi-3\beta)}{\sin(\beta)}\right)$$</span> and with... |
1,065,988 | <p>In <a href="https://www.youtube.com/watch?v=6Lm9EHhbJAY">this Numberphile video</a> it is stated that trisecting an angle is not possible with only a compass and a straight edge. Here's a way I came up with:</p>
<blockquote>
<p>Let the top line be A and bottom line be B, and the point intersecting P.<br>
1. Use... | Victor Liu | 398 | <p>Let your angle be almost 180 degrees, so your two lines are almost coincident. The line $MN$ is also almost parallel to the lines, and trisecting that segment leads to very unequal "thirds" of the angle.</p>
|
1,065,988 | <p>In <a href="https://www.youtube.com/watch?v=6Lm9EHhbJAY">this Numberphile video</a> it is stated that trisecting an angle is not possible with only a compass and a straight edge. Here's a way I came up with:</p>
<blockquote>
<p>Let the top line be A and bottom line be B, and the point intersecting P.<br>
1. Use... | Blue | 409 | <p>Let $X$ and $Y$ be the two points you added to $\overline{MN}$, as shown. </p>
<p><img src="https://i.stack.imgur.com/zIeiIm.png" alt="enter image description here"></p>
<p>Consider $\triangle PMY$. Assuming your trisection construction to be valid, $\overline{PX}$ must <em>bisect</em> $\angle MPY$. By the <a href... |
4,578,898 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$f$</span> be the function defined on <span class="math-container">$[0,1]$</span> by
<span class="math-container">$$
f(x)=
\begin{cases}
n(-1)^n & \textrm{if }\frac{1}{n+1}<x\leq \frac{1}{n} \\
0 & \textrm{otherwise}
\en... | Ross Millikan | 1,827 | <p>In math we define the <span class="math-container">$n^{th}$</span> root of <span class="math-container">$k$</span> to be the number that when multiplied by itself <span class="math-container">$n$</span> times results in <span class="math-container">$k$</span>. We write this <span class="math-container">$m=\sqrt[n]k... |
461,268 | <p>A square hole of depth $h$ whose base is of length $a$ is given.
A dog is tied to the center of the square at the bottom of the hole by a rope of length
$L>\sqrt{2a^2+h^2}$ ,and walks on the ground around the hole.</p>
<p>The edges of the hole are smooth, so that the rope can freely slide along it. Find the sha... | hasnohat | 46,166 | <p>I tried a simplified version of this problem, where the dog is facing a cliff of height $h$, which I represent as the line $y=\frac{a}{2}$. In other words, considering only one side of the square, and extending it to form a line.</p>
<p>After a bunch of hastily scribbled algebra, I arrived at the following curve de... |
2,871,947 | <p>I can prove that commutator is minimal subgroup such that factor group of it is abelian. I had encountered one statement as</p>
<blockquote>
<p>If <span class="math-container">$H$</span> is a subgroup containing commutator subgroup then <span class="math-container">$H$</span> is
normal.</p>
</blockquote>
<p>I.... | José Carlos Santos | 446,262 | <p>If $g\in G$ and $h\in H$, then $ghg^{-1}h^{-1}=h'$, for some $h'\in H$ (since $H$ contains the commutator subgroup). But then $ghg^{-1}=h'h\in H$. Therefore, $gHg^{-1}\subset H$.</p>
|
85,984 | <p>I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic geometry.</p>
<p>I am considering doing research in mathematics to be my career (and my life) someday. and I am wondering ab... | Joseph Malkevitch | 1,369 | <p>You can find many papers about recent work in advanced "classical" Euclidean geometry at this site: <a href="http://forumgeom.fau.edu/">http://forumgeom.fau.edu/</a> </p>
|
26,420 | <p>So the cup product is not well defined over co-chain groups, but all the books claim it is well defined over co-homology groups. The only thing I am not clear on is invariance under ordering/re-ordering of simplices when we go to the co-homology level. Every book seems to gloss over this, and after doing a few examp... | Akhil Mathew | 536 | <p>Strictly speaking, the cup product is not commutative, though it is commutative up to sign on the level of cohomology.</p>
<p>There is an abstract way of seeing this: namely, we can use the method of acyclic models. Consider the following two functors from the category of spaces to the category of chain complexes.... |
26,420 | <p>So the cup product is not well defined over co-chain groups, but all the books claim it is well defined over co-homology groups. The only thing I am not clear on is invariance under ordering/re-ordering of simplices when we go to the co-homology level. Every book seems to gloss over this, and after doing a few examp... | rhl | 8,122 | <p>right, ok. I asked about if cup product is well defined, here is why I am confused.</p>
<p>You wrote:</p>
<hr />
<p>Strictly speaking, the cup product is not commutative, though it is commutative up to sign on the level of cohomology.</p>
<p>There is an abstract way of seeing this...</p>
<hr />
<p>The cup product be... |
706,250 | <p>For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$.</p>
<p>I tried to set $n = 3+4k$ but it doesn't help.</p>
<p>Any hints first please?</p>
| DeepSea | 101,504 | <p>Write $n = 4k + 3 \implies 3^n = 3^{(4k + 3)} = 27*81^k$. We know $27 \equiv 2 (\mod 5)$, and $81 \equiv 1 (\mod 5)$ . So $81^k \equiv 1^k = 1 (\mod 5)$. So $3^n \equiv 2*1 = 2 (\mod 5)$.</p>
|
706,250 | <p>For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$.</p>
<p>I tried to set $n = 3+4k$ but it doesn't help.</p>
<p>Any hints first please?</p>
| Community | -1 | <p>This is a perfect chance for induction: If $n = 3$, this is quite clear. Now suppose that the claim is true for $n = 4k + 3$. We then have</p>
<p>\begin{align*}
3^{4(k + 1) + 3} &= 3^{4k + 3} 3^4
\end{align*}</p>
<p>Now $3^4 = 3^{5 - 1}$, and Fermat's Little Theorem is applicable (or just write down $3^4$ mod ... |
262,509 | <p>I have a list of data <code>{{x1,c11}, {x1,c12}, {x2,c21},{x2,c22}, {x3,c31},{x3,c32}, {x4,c4}, {x5,c51},{x5,c52}, ...}</code>, in which <code>x</code> is real value and <code>c</code> is complex value. For some <code>x</code>, there is only a <code>c</code> value, for example, <code>{x4,c4}</code> pair, while for t... | kglr | 125 | <pre><code>test2 = test /. Complex[a_, b_] :> {a, b};
</code></pre>
<p><strong>1.</strong></p>
<pre><code>{xim, xre} = Transpose[Transpose /@
GatherBy[Reverse @* Thread /@ test2, Sign[.4 + First @ # ] &]];
Row[ListPlot[#, ImageSize -> 400, PlotRange -> All ] & /@ {xim, xre}, Spacer[20]]
</code></... |
557,680 | <blockquote>
<p>Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that $AA^t=I_k$. Let $f:M_{k,n}\rightarrow S_k$ be the map $f(B)=BA^t+AB^t$. Prove that $f$ is onto (surjective). (Not... | Sammy Black | 6,509 | <p>Let $B = \frac{1}{2}CA \in M_{k, n}$.</p>
|
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