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 |
|---|---|---|---|---|
31,451 | <p>I have created a notebook, with functions. The only thing that I want to change in the notebook later is the input file. I therefor want to have a button that runs every input step-by-step so that I do not have to go into every input and press enter for it to update. How could this be done. </p>
| bill s | 1,783 | <p>Perhaps the simplest thing is to command-A and enter, which selects all the cells and evaluates. Or you can use a button</p>
<pre><code>evalStart := Module[{},
FrontEndExecute[FrontEndToken[InputNotebook[], "EvaluateInitialization"]];
FrontEndExecute[FrontEndToken[InputNotebook[], "SelectAll"]];
FrontE... |
1,074,809 | <p>What is a non-decreasing sequence of sets and how come it can have a limit?</p>
<p>It appear in a probability theory book</p>
| Swapnil Tripathi | 117,387 | <p><strong>Hint:</strong> $\sqrt{y}$ is defined for $y\ge 0$. So $2x-1\ge 0$ is the domain of the function. (as $\sin$ is defined on whole $\mathbb{R}$)</p>
|
325,860 | <p>Let $a,b \in \mathbb{R}$, $a<b$ and let $f$ be a differentiable real-valued function on an open subset of $\mathbb{R}$ that contains [a,b]. Show that if $\gamma$ is any real number between $f'(a)$ and $f'(b)$ then there exists a number $c\in(a,b)$ such that $\gamma=f'(c)$.</p>
<p>Hint: Combine mean value theore... | Pedro M. | 21,628 | <p>Hint: Let $$g(x_1,x_2) = \frac{f(x_1) - f(x_2)}{x_1 - x_2}$$</p>
<p>Then $$\lim_{x_2 \searrow a} g(a,x_2) = f'(a) \text{ and } \lim_{x_1 \nearrow b} g(x_1,b) = f'(b)$$ </p>
<p>So if $\varepsilon$ is sufficiently small, what can you say about $g(a,a+\varepsilon), \gamma$ and $g(b-\varepsilon,b)$?</p>
<p>Now if you... |
2,640,909 | <p>I encountered a problem with 4 variables and I was wondering if anyone knows how to solve this:
This is what is known:</p>
<p>$$ \left\lbrace
\begin{align}
a+b &= 1800 \\
c+d &= 12 \\
a/c &= 100 \\
b/d &= 250 \\
(a+b)/(c+d) &= 150
\end{align}
\right.$$</p>
<p>Below is a screenshot from a spread... | Yash Jain | 522,158 | <p>$b=1800-a$</p>
<p>$d=12-c$</p>
<p>$\cfrac ac=100$</p>
<p>$a=100c$</p>
<p>$\cfrac {1800-a}{12-c}=250$</p>
<p>$\cfrac {2.5a}c=250$</p>
<p>$\cfrac {1800-a}{12-c}=\cfrac {2.5a}c$</p>
<p>$\cfrac {1800-100c}{12-c}=\cfrac {2.5(100c)}c$</p>
<p>$1800c-10c^2=250c(12-c)$</p>
<p>Which gives you the value of $c$</p>
|
342,491 | <p>How to prove the following:</p>
<p>$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$</p>
| Aryabhata | 1,102 | <p>There was a comment that you can use $\text{AM} \ge \text{GM}$ to prove the boundedness of this.</p>
<p>Here is a proof. As a side effect, we also prove the convergence.</p>
<p>First we show that $x_n = \left(1 + \frac{1}{n}\right)^n$ is monotonically increasing. We prove this, using $\text{AM} \ge \text{GM}$.</p>... |
58,710 | <p>I've got a bunch of readings at various dates and times. I'd like to be able to interpolate and then integrate over, say days. I can do this by converting my date/time to <code>AbosluteTime</code> and get the interpolating function that way. Then by integrating and plugging in seconds as my limits of integration ... | Dr. belisarius | 193 | <p>Some sample data: </p>
<pre><code>r = # - #[[1]] &@(Range[##, 3690] & @@ (AbsoluteTime[{#, {"Day", "Month", "YearShort"}}] & /@
{"05/01/14", "05/03/14"}));
hours = N[r/3600];
data = Transpose[{hours, hours^2}];
</code></pre>
<p>... |
505,067 | <p>I am writing a computer program which generate 5 digit alpha-numeric codes. Each character can contain a-z,A-Z and 0-9 (62 different possibility per character).</p>
<p>So if there are 5 character how many possibility are there?</p>
| Old John | 32,441 | <p>If it is not an integer, there must be some prime which divides the denominator but not the numerator. The same will be true of all integer powers $n\ge 1$, so it cannot be an integer.</p>
|
2,442,233 | <p>let $A= \{x^2 \mid 0 < x <1\}$ and $B =\{x^3 \mid 1 < x < 2 \}$.</p>
<p>Which of the statement is true?</p>
<p>1.there is a one to one, onto function from $A$ to $B$.</p>
<p>2.there is no onto function from $A$ to $B$</p>
<p>my attempt ; there will be no onto function from $A$ to $B$ because order ... | operatorerror | 210,391 | <p>Note that
$$
B=(1,2^3)=(1,8)\\
A=(0,1)
$$
Then, take
$$
f:A\to B\\
f(x)=7x+1
$$
which is linear and thus invertible. </p>
|
3,497,554 | <blockquote>
<p>Show that <span class="math-container">$\frac{(2n-1)!}{(n)!(n-1)!}$</span> is odd or even according as <span class="math-container">$n$</span> is or is not a power of <span class="math-container">$2$</span>.</p>
</blockquote>
<p>I know that the index of the highest power of <span class="math-containe... | ViHdzP | 718,671 | <p>Let <span class="math-container">$\nu_2(x)$</span> be the exponent of the highest power of <span class="math-container">$2$</span> that divides <span class="math-container">$x$</span>. By <a href="https://en.wikipedia.org/wiki/Legendre's_formula" rel="nofollow noreferrer">Legendre’s Formula</a>, </p>
<p><span c... |
830,755 | <p>I would like to compute the following,
$$
\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy
$$
It is obvious that we can rewrite the integral above to,
$$
\int_0^{\infty}\int_0^{\infty}e^{-(x+y)^2}\ dx\,dy
$$
so we are ending up with something looking like a gaussian integral. I think that a smart substitution ... | Pranav Arora | 117,767 | <p>Switch to polar coordinates i.e $x=r\cos\theta$, $y=r\sin\theta$ and $dx\,dy=r\,dr\,d\theta$ to obtain:
$$\int_0^{\pi/2} \int_0^{\infty} re^{-r^2(1+\sin(2\theta))}dr\,d\theta=\int_0^{\pi/2} \frac{1}{2(1+\sin(2\theta))}\,d\theta$$
Write $\sin(2\theta)=\frac{2\tan\theta}{1+\tan^2\theta}$ to get:
$$\int_0^{\pi/2} \frac... |
2,190,508 | <p>Let $f:[a,b] \to \mathbb R$ be nice differentiable function. By Fundamental Theorem of Calculus (FTC), we have $$\int_a^{b} f'(t) dt = f(b) -f(a).$$</p>
<p>By Cauchy-Schwartz inequality, we have, $\int_{a}^{b} |f(t)| dt \leq (b-a)^{1/2}(\int_a^b |f(t)|^2 dt)^{1/2} .$ Using this, and FTC, we have $\int_{a}^{b} (... | Micah | 30,836 | <p>In answer to (1), let $f(x)=e^{x^2/2}$. Then $(f'(x))^2=(xe^{x^2/2})^2=x^2e^{x^2}$. This has no elementary antiderivative, as we have
$$
\int x^2e^{x^2} \, dx=\int x(xe^{x^2} \, dx)= \frac{1}{2}xe^{x^2}-\frac{1}{2}\int e^{x^2}\, dx
$$
and this last integral is well-known not to be elementary.</p>
<p>It follows that... |
3,810,944 | <p>Consider a positive integer <span class="math-container">$n$</span> and the function
<span class="math-container">$f:\mathbb{N}\to \mathbb{N}$</span> (<span class="math-container">$\mathbb N$</span> includes <span class="math-container">$0$</span>) by</p>
<p><span class="math-container">$$f(x) = \begin{cases} \fra... | Community | -1 | <p><strong>Hint:</strong></p>
<p>Letting all divisions be integer and splitting the cases after the low order bits of <span class="math-container">$x$</span>, we can expand the function iterates for, say, <span class="math-container">$n=4$</span>, and the pattern becomes obvious.</p>
<p><span class="math-container">$$f... |
875,924 | <p>For what values of $x\in\mathbb{R}$ is $f$ continuous?</p>
<p>$f(x) = \left\{
\begin{array}{lr}
0 & \text{if}\, x \in \Bbb Q\\
1 & \text{if}\, x \notin \Bbb Q
\end{array}
\right.$</p>
<p>The solution I found:</p>
<p>$f(x)$ is continuous nowhere. For, given any number $a$ and any... | Mhenni Benghorbal | 35,472 | <p>You can use <a href="http://www.math.washington.edu/~morrow/334_13/continuity.pdf" rel="nofollow">sequential continuity</a>.</p>
|
322,448 | <p>I'm supposed to show that: </p>
<p>$$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ </p>
<p>and the required answers are: $$ P=5, Q=4, R=-4 $$ </p>
<p>I tried to solve this with partial fractions like so: </p>
<p>$$5(x-1)(x+2) = A(x+3) + B(x-2)$$ </p>
<p>$\implies$ $A$=4, $B$=-4... | Community | -1 | <p>$$\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ gives<br>
$$5(x-1)(x+2)=P(x-2)(x+3)+Q(x+3)+R(x-2)........(1)$$<br>
Now put $x=2, x=-3,x=0$ respectively on both sides of $(1)$ to get, $Q=4,R=-4,P=5$ respectively. Hence you can reach the desired result.</p>
|
2,207,260 | <p>I have to find the asymptotics of the following integral</p>
<p>$$\int^1_0 \frac{\sin(x)}{x(1+x)^n} dx$$</p>
<p>as $n\to\infty$.</p>
<p>I know I am supposed to use the lebesgue dominated convergence theorem and create a sequence $b_n$ and aim for $\lim_{n\to\infty} a_n/b_n =1$ but I can't seem to get my head roun... | tired | 101,233 | <p>Denote the integral in question by $I_n$ and squeeze it baby!</p>
<p><a href="https://math.stackexchange.com/questions/390899/prove-that-x-fracx33-sin-x-x-for-all-x0">Notice</a> that the inequalities $x-x^3/3!<\sin(x)<x$ hold on $x\in(0,1)$</p>
<p>so</p>
<p>$$
\int_0^1dx\frac{1-x^2/3!}{(1+x)^n}<I_n<
\... |
882,540 | <p>Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities?</p>
<ol>
<li><p>$(x+a)-a=x$</p></li>
<li><p>$(x-a)+a=x$</p></li>
<li><p>$(x+y)+a = (x+a)+(y+a)$</p></li>
<li><p>$(x-y)+a = (x+a)-(y+a)$</p></li>
</ol>
<p><em>Remark.</em> The Abelian gro... | Jack Yoon | 141,253 | <p>Well I get one stupid example where $X$ is any set and $+$ and $-$ act trivially. What I mean by trivially is:</p>
<p>$x+a=x-a=x$ for all $x, a \in X$. Which is clearly not interesting.</p>
|
102,280 | <p>What are the usual tricks is proving that a group is not simple? (Perhaps a link to a list?)</p>
<p>Also, I may well be being stupid, but why if the number of Sylow p groups $n_p=1$ then we have a normal subgroup?</p>
| edgar | 23,702 | <p>The summation
$$ \sum\limits_{i=0}^n \frac{i}{2^{i}} $$</p>
<p>is equal to
$$ \sum\limits_{i=0}^n i(\frac{1}{2})^i $$</p>
<p>which is of the form
$$ \sum\limits_{i=0}^n ix^i, x = \frac{1}{2} $$</p>
<p>Consider the summation $$ \sum\limits_{i=0}^n x^i = \frac{x^{n+1} - 1}{x - 1} $$</p>
<p>By differentiating wit... |
3,374,024 | <p>Basicly, i am bad at mathematics very hard to me got score 70. I try learn trigonometry now, starting from the very basic.
I practice to find the value of <span class="math-container">$$\sin(135^\circ)$$</span> using my knowing about <span class="math-container">$\sin(45^\circ)={\sqrt2\over2}$</span>, <span class="m... | Taroccoesbrocco | 288,417 | <p>'True' refers to the English statement (2.2), 'correct' refers to the sequent <span class="math-container">$\Gamma \vdash \Psi$</span>. The authors aim to define when a sequent is <em>correct</em> in a technical sense, and they use the common sense of 'true' in natural language (the meta-level).</p>
<p>In other wor... |
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | lhf | 589 | <p>What you're probably missing is that, although the real cube root of 2 stands out, <em>all</em> three complex cube roots of 2 are algebraically the same, that is, they cannot be told apart, and so can be permuted at will. Hence $S_3$.</p>
<p>Contrast this with the seemingly analogous situation of, say, the 6-th roo... |
1,939,107 | <p>An unbiased coin is tossed repeatedly and outcomes are recorded. What is the expected no of toss to get HT ( one head and one tail consecutively) ?</p>
<p>I tried to solve above question using the following probability tree diagram.</p>
<p><a href="https://i.stack.imgur.com/10cO3.png" rel="nofollow noreferrer"><im... | JMoravitz | 179,297 | <p>You can treat this as a <a href="https://en.wikipedia.org/wiki/Markov_chain" rel="nofollow">Markov Chain</a> with the following states:</p>
<p>End, Head, Tail (not end) (<em>or equivalently worded, HT has occured, heads is most recent result, and no heads have occured respectively</em>)</p>
<p>This has the transi... |
3,238,306 | <p>Hello I have the differential equation <span class="math-container">$$x'=\frac{8t+10x}{17t+x}$$</span>
Brought it in a eqation where I can substitute <span class="math-container">$$ u=\frac{x}{t}$$</span>
and after some transformations I got the equation
<span class="math-container">$$ \frac{17+u}{8-7u-u^2}du=\frac... | Community | -1 | <p>Let <span class="math-container">$a\equiv x+ 2,\,b\equiv y+ 2$</span>
<span class="math-container">$$\therefore\,ab- a- b= xy+ x+ y\geqq 0$$</span>
<span class="math-container">$$\because\,x,\,y\geqq 0$$</span></p>
|
290,173 | <p>Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\widetilde{d}(x,y)\to\infty$ if and only if $d(x,y)\to K$? One way would be to find some function $f$ such that $\widetil... | Adrián González Pérez | 12,604 | <p>I do not know of any procedure for constructing such metric space in the general setting. On the contrary, if you restrict your class of metric spaces to the smaller class of complete locally compact path-metric spaces then the Hopf-Rinow theorem, see the First Chapter of [Gr], will prevent the construction of your ... |
114,340 | <p>Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've
$$\text{ rank of }\left(
\begin{array}{ccc}
0 &\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w} \\
\frac{\partial F}{\partial z} &\frac{\partial^2 F}{\partial ^ 2z} &\frac{\... | Qiaochu Yuan | 232 | <p>The story extends to non-square matrices. Consider a collection $A$ of $a$ vertices, another collection $B$ of $b$ vertices, and a collection of directed edges from the first collection to the second (in other words, a bipartite graph, but where we choose explicitly a $2$-coloring of the vertices). We can organize t... |
1,359,743 | <p>If $\tan \theta = 3\cfrac{15}{16}$, then find $\sin \theta$.</p>
| Rick | 251,424 | <p>Let us rewrite $3 \frac{15}{16}$ as an improper fraction. So we get $3 \dfrac{15}{16} = \dfrac{16\cdot3+15}{16} = \dfrac{63}{16}$</p>
<p>We know that $\tan(\theta) = \dfrac{o}{a}$, where $o$ and $a$ are the opposite and adjacent sides of $\theta$. We also know through the Pythagorean theorem that the hypotenuse is ... |
1,403,228 | <blockquote>
<p><strong>Question:</strong></p>
<p>Let <span class="math-container">$m, n, q, r \in \mathbb Z$</span>. If <span class="math-container">$m = qn + r$</span>, show that <span class="math-container">$\gcd(m, n) = \gcd(n, r)$</span>. Hence justify the Euclidean Algorithm.</p>
</blockquote>
<p>I found this que... | Goob | 953,211 | <p>Let <span class="math-container">$m=qn+r$</span>. We have <span class="math-container">$gcd(m,n)=gcd(qn+r,n)=gcd(r,n)$</span>, where the last equality follows that if <span class="math-container">$d=gcd(qn+r,n)$</span> then since <span class="math-container">$d|n$</span> we know <span class="math-container">$d|qn$</... |
2,871,084 | <p>Consider that we have a circle drawn a round the <strong>origin (0,0)</strong>. That circle has some points drawn on its circumference. Each of those points has range and azimuth $(r,\theta)$, the <strong>r</strong> and <strong>$\theta$</strong> values of these points are calculate with responding to the <strong>ori... | Batominovski | 72,152 | <p>Assume that $\mathcal{C}\big([a,b]\big)$ is taken modulo constant functions. Any function in the image of $R$ is differentiable everywhere except on a set of Lebesgue measure $0$, but not every continuous function has this property. A properly shifted and rescaled version of Weierstrass function as suggested by Qu... |
3,170,368 | <p>The limit of <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span> as <span class="math-container">$(x,y) \longrightarrow(0,0)$</span> is doesn't exist, because if we take two paths:</p>
<p>1) Along the path, <span class="math-container">$x=0$</span>, <span class="math-container">$y\longrightarrow0$</sp... | Kavi Rama Murthy | 142,385 | <p>When you fixing <span class="math-container">$\theta$</span> and take limit as <span class="math-container">$ r \to 0$</span> you are taking limits along straight lines. It is not enough to take limits along these lines. When you take limit along the parabola <span class="math-container">$x=y^{2}$</span> we get a d... |
3,170,368 | <p>The limit of <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span> as <span class="math-container">$(x,y) \longrightarrow(0,0)$</span> is doesn't exist, because if we take two paths:</p>
<p>1) Along the path, <span class="math-container">$x=0$</span>, <span class="math-container">$y\longrightarrow0$</sp... | Dagnachew Jenber | 627,123 | <p>I asked this question a couple of days ago. I want to answer my question myself, I'm not fully satisfied with your answers. Let me get started by stating the following Theorem with out proof. </p>
<p><strong>Theorem 1.</strong> If the limits of <span class="math-container">$f(x,y)$</span> and <span class="math-cont... |
1,870,675 | <blockquote>
<p>Let $R$ be a commutative ring with unity, and let $R^{\times}$ be the group of units of $R$. Then is it true that $(R,+)$ and $(R^{\times},\ \cdot)$ are not isomorphic as groups ? </p>
</blockquote>
<p>I know that the statement is true in general for fields. And it is trivially true for any finite ri... | arctic tern | 296,782 | <p>Counterexample: $R=\mathbb{R}\times\mathbb{Z}_2$ satisfies $(R,+)\cong(R^\times,\cdot)$.</p>
|
221,403 | <p>I have a series of functions <span class="math-container">$f_d$</span> in <span class="math-container">$d$</span> variables and would like compute the sum of each one evaluated at each lattice point within the <span class="math-container">$d$</span>-sphere of radius <span class="math-container">$R$</span>; that is, ... | Vitaliy Kaurov | 13 | <h2>Method 1: select after building the grid</h2>
<p>The cubic lattice with side size <span class="math-container">$2r$</span> in <span class="math-container">$d$</span>-dimensions symmetric wrt origin can be defined as:</p>
<pre><code>latt[r_,d_]:=Tuples[Range[-r,r],d]
</code></pre>
<p>Then your points are selected... |
1,450,669 | <p>Find the value of $$\lim\limits_{x\to 0^+}[1+[x]]^{\frac2x}$$where $[x]$ denotes greatest integer function less than or equal to $x$. <br></p>
<hr>
<p><strong>My attempt:</strong> <br>
I calculated $[1+[x]]$ to be $1$ as $x\to 0^+$. <br> Now I am stuck. <br> Please help me.</p>
| Calvin Khor | 80,734 | <p>For $0<x\ll 1$, $[1+[x]] = 1$. Thus $[1+[x]]^{2/x} = 1^{2/x} = 1$.This gives
$$ \lim_{x→ 0^+}[1+[x]]^{2/x} = \lim_{x→ 0^+} 1 = 1$$</p>
|
3,493,729 | <p>Although I looked up the answer on integral calculator com but I still have little to no idea as to how one would proceed to solve this integral.
Integrate <span class="math-container">$\dfrac{e^x(x^4+2)}{(1+x^2)^{5/2}}$</span> wrt <span class="math-container">$x.$</span> I initially tried to convert it to the form ... | Kavi Rama Murthy | 142,385 | <p>If it is onto then <span class="math-container">$f^{-1}(0)$</span> and <span class="math-container">$f^{-1}(1)$</span> are disjoint closed non-empty subsets of <span class="math-container">$[0,1]$</span> which contradicts connectedness of <span class="math-container">$[0,1]$</span>. Hence such a function does not ex... |
3,425,780 | <p>Let <span class="math-container">$H_i$</span> be a <span class="math-container">$\mathbb C$</span>-Hilbert space and <span class="math-container">$T$</span> be a densely-defined linear operator from <span class="math-container">$H_1$</span> to <span class="math-container">$H_2$</span>.</p>
<blockquote>
<p>How can... | Toby Mak | 285,313 | <p>The equation of the parabola is:</p>
<p><span class="math-container">$$a \left(x-\frac{1}{4}\right)^2 - \frac{9}{8}$$</span>
<span class="math-container">$$=a \left(x^2-\frac{1}{2}x+\frac{1}{16} \right)- \frac{9}{8}$$</span>
<span class="math-container">$$=ax^2-\frac{a}{2}x+ \left(\frac{a}{16}-\frac{9}{8} \right)$$... |
2,341,817 | <blockquote>
<p>If <span class="math-container">$ \log_p q + \log_q r + \log_r p = 0 $</span></p>
<p>Then what is the value of,</p>
<p><span class="math-container">$$(\log_p q)^3 + (\log_q r)^3 + (\log_r p)^3$$</span></p>
<p>given that <span class="math-container">$p,q,r \neq 1$</span></p>
<p>A. It is odd prime</p>
<p... | Tianlalu | 394,456 | <p><strong>HINT</strong></p>
<p><br>
$x^3+y^3+z^3=3xyz$ when $x+y+z=0$.</p>
<p>AND</p>
<p><br>
$\log_p q=\frac{\log q}{\log p}$, $\log_q r=\frac{\log r}{\log q}$, $\log_r p=\frac{\log p}{\log r}$.</p>
|
2,341,817 | <blockquote>
<p>If <span class="math-container">$ \log_p q + \log_q r + \log_r p = 0 $</span></p>
<p>Then what is the value of,</p>
<p><span class="math-container">$$(\log_p q)^3 + (\log_q r)^3 + (\log_r p)^3$$</span></p>
<p>given that <span class="math-container">$p,q,r \neq 1$</span></p>
<p>A. It is odd prime</p>
<p... | John D | 389,906 | <p>Based on the identity </p>
<p>$$a+b+c=0 \Rightarrow a^3+b^3+c^3 =3abc,$$ the answer is 3. Just note that </p>
<p>$\log_p(q)=\frac{ln(q)}{ln(p)},$ etc, so your hypothesis is just </p>
<p>$$\frac{ln(q)}{ln(p)}+\frac{ln(r)}{ln(q)}+\frac{ln(p)}{ln(r)}=0.$$ Put now </p>
<p>$$a= \frac{ln(q)}{ln(p)} , \; b=\frac{ln(r)}... |
1,214,042 | <p>"A particle of mass m is attracted towards a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at a distance L from 0, find the time for it to reach 0"</p>
<p>My attempt at this question:
Noting the fact $x(0)=L$ and $x'(0)=0$, I tried to ... | JJacquelin | 108,514 | <p>$$2m\frac {d^2x}{dt^2}\frac {dx}{dt}=\frac {-2k}{x}\frac {dx}{dt}$$
$$m(x')^2=-2k\ln(x)+constant$$
With conditions $x(0)=L$ and $x'(0)=0$ :
$$m(x')^2=-2k\ln(x)+2k\ln(L)$$
$x$ is decreassing, hense $x'\leq 0$
$$x'=-\sqrt{-\frac{2k}{m}\ln{\frac{x}{L}}}$$
$$\frac{dx}{\sqrt{\ln(L)-\ln(x)}}=-\sqrt{\frac{2k}{m}}dt$$
$$\in... |
2,462,852 | <p>I want to prove that $f(x)=x^3+x+2$, $f: \mathbb R \rightarrow \mathbb R$ is bijective without calculus. My attempts at showing to prove that it' injective and surjective are written below:</p>
<p>$1)$ Injectivity: </p>
<p>I want to show that $\forall a,b \in \mathbb R$ $f(a)=f(b) \implies a=b$.</p>
<p>I started ... | Piquito | 219,998 | <p>Injectivity is clear and so is for the fact that there is not a positive root. </p>
<p>Applying now Descartes's Rule we have
$$f(x)=x^3+x+2 \text{ have no change signs }\\f(-x)=-x^3-x+2 \text{ has one change sign }$$
Consequently the maximum number of negative roots is $0+1=1$ and $f^{-1}$ exists.</p>
|
284,995 | <p>There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode.</p>
<p>May I know if the converse is true?</p>
<p>(i.e. if $S^2=id_H$, does it follow that $H$ is commutative or cocommutative?)</p>
<p>If no, what would be the simplest counter-example?... | Bogdan | 13,737 | <p>The converse is not true, Sweedler's 4-dimensional Hopf algebra (first example of a non-comm and non-cocomm Hopf algebra) is a counter example. This example is completely taken from section 4.3.6 of "Hopf Algebras" by Dascalescu, Nastasescu and Raianu. </p>
<p>Let $k$ be a field of characteristic $\neq 2$. As a vec... |
165,235 | <p>I have a weighted, directed graph with 100 vertices and the maximal number of edges, 9900. Are there <em>Mathematica</em> tools or packages available to visualize which edges have large weights? (If you know of non-<em>Mathematica</em> software, that's fine, too.)</p>
<p>Background: My institution has 100 majors... | Tugrul Temel | 60,365 | <p>To draw the subgraph with large weights, you might want to apply the following function:</p>
<pre><code>select[matrix_, lB_, uB_] :=
matrix*Map[Boole[lB <= # <= uB] &, matrix, {-1}];
sa = SparseArray[select[mat, .1, .6]];
weightedG = Graph[sa["NonzeroPositions"], EdgeWeight -> sa["Nonz... |
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | nonuser | 463,553 | <p>Suppose it is a polynomial, then <span class="math-container">$$\sqrt{1+x^2}= ax^n+...+b$$</span></p>
<p>so <span class="math-container">$$1+x^2= (ax^n+...+b)^2 = a^2x^{2n}+...$$</span></p>
<p>Since polynomial are equal iff they have same degree we have <span class="math-container">$2=2n$</span> so <span class="ma... |
2,846,700 | <p>Consider the functions $f(x)$, $g(x)$, $h(x)$, where $f(x)$ is neither odd nor even, $g(x)$ is even and $h(x)$ is odd. Is it possible for $f(x) + g(x)$ to be </p>
<ol>
<li>even;</li>
<li>odd?</li>
</ol>
<p>For the second case I can imagine for example $f(x) = x - 1$ and $g(x) = 1$. Then $f$ is neither even nor od... | Cato | 357,838 | <p>f(x) = f(-x)</p>
<p>$g(x) \ne g(-x)$ for some x</p>
<p>z(x) = f(x) + g(x)</p>
<p>then $z(-x) = f(-x) + g(-x) \ne f(x) + g(x) $ for some x</p>
|
1,718,391 | <p>The sequence is </p>
<p>$$\sum_{n=1}^{\infty} \frac{n-1}{n!}$$</p>
<p>I could show that it converges using the Ratio test, but evaluating it seems to be hard. I'm trying to avoid power series and Taylor series and such. Is it possible to solve this some other, simpler way? </p>
<p>I tried writing out the first fe... | DonAntonio | 31,254 | <p>Your sum is only</p>
<p>$$\sum_{n=1}^\infty\frac{n-1}{n!}=\sum_{\color{red}{n=0}}^\infty\overbrace{\frac1{n!}}^{=\frac n{n!}=\frac1{(n-1)!}}-\sum_{\color{red}{n=1}}^\infty\frac1{n!}=e^1-(e^1-1)=1$$</p>
|
701,110 | <p>A particular combinatorial sequence I was looking at turned out to obey the following pair of recurrence relations:</p>
<p>$$N_{2n+1}=\sum^n_{k=0}N_{2k}$$
$$N_{2n}=\left(\sum^{n-1}_{k=0}N_{2k+1}\right)+1=\sum^n_{k=0}N_{2k-1}$$</p>
<p>For the second form of the second equation, I'm using the convention $N_{-1}=1$.<... | Ross Millikan | 1,827 | <p>Yes, this is preserved as we go to higher dimensions as long as the number of terms is the same as the dimension of the space. The Pythagorean theorem (iterated) says this sum of squares is the square of the distance from the origin to the point.</p>
|
701,110 | <p>A particular combinatorial sequence I was looking at turned out to obey the following pair of recurrence relations:</p>
<p>$$N_{2n+1}=\sum^n_{k=0}N_{2k}$$
$$N_{2n}=\left(\sum^{n-1}_{k=0}N_{2k+1}\right)+1=\sum^n_{k=0}N_{2k-1}$$</p>
<p>For the second form of the second equation, I'm using the convention $N_{-1}=1$.<... | copper.hat | 27,978 | <p>Sure, if $Q$ is a rotation matrix ($Q^T Q = I$), then with the Euclidean norm we have
$\|x\|^2 = \langle x , x \rangle = \langle x , Q^T Qx \rangle = \langle Qx , Qx \rangle = \|Qx\|^2$.</p>
|
873,224 | <p>I'm not mathematically inclined, so please be patient with my question.</p>
<p>Given </p>
<ul>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ as the endpoints of a cubic Bezier curve.</p></li>
<li><p>$(c_x, c_y)$ and r as the centerpoint and the radius of a circle.</p></li>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ are on the ci... | me22 | 875,771 | <p>As a different way to choose where the error falls, one could pick the control points such that the <em>enclosed area</em> matches that of the circle.</p>
<p>If we consider the first quarter of a circle (the same as the first image in the other answer), then the area should be <span class="math-container">$\pi/4$</s... |
1,390,593 | <p>"Since the operation of left multiplication is faithful, $G$ is isomorphic to its image in $\operatorname{Perm}(G)$. If $G$ has order $n$, $\operatorname{Perm}(G)$ is isomorphic to $S_n$." - Artin's Algebra</p>
<p>So left multiplication as an action means $\phi_g(x)=gx$</p>
<p>So proving isomorphism, means bijecti... | mich95 | 229,072 | <p>$\phi(g)$ is bijective as you said, so it belongs to $Perm(G) \cong S_{n}$. Now the map, used in Cayley's theorem, is $f:G \to Perm(G)$, $f(g) =\phi_{g}$. $\phi$ is not necessarly a homomorphism, but $f$ is a homomorphism, in fact injective. Let me know if you need more details!</p>
|
3,166,201 | <blockquote>
<p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are prime numbers which satisfy <span class="math-container">$x^2-2y^2=1$</span>, solve for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</blockquote>
<p><strong>My attem... | Mann | 126,204 | <p>The fault is that irrationals can also produce the product to <span class="math-container">$1$</span>.<br>
Consider <span class="math-container">$x=3$</span> and <span class="math-container">$y=2$</span> then we get, <span class="math-container">$(3+\sqrt{2} \cdot 2)(3-\sqrt{2}\cdot 2)=1$</span><br>
Hence, the fault... |
1,897,163 | <p>Let $\frac{x^2}{y}\to c\neq 0$ as $(x,y)\to (0,0)^+$. Why does this mean that $y$ goes faster to $0$ as $x$ does?</p>
<p>For values of x and y that are small enough, we surely do not have $x^2/y=1$ so they cannot be identical, okay. Moreover, if the quotient would tend to 0, this would mean that $x$ would be faster... | guestDiego | 338,527 | <p>Of course $f_n(0)=0$ for any $n$ and there is nothing to prove. Suppose $x\neq 0$. Then $1+x^2=:q>1$ and
$$
\lim_{n\to +\infty} \frac{1}{q^n}=0,
$$
therefore
$$
\lim_{n\to +\infty}f_n(x)=x\cdot \lim_{n\to +\infty} \frac{1}{q^n}=x\cdot 0=0,
$$</p>
|
3,575,015 | <p>This question is taken out of the text book High Dimensional Probability by Roman. I am unsure from the context if it means locally or globally Lipschitz. But the question is as follows:</p>
<p>Every differentiable function <span class="math-container">$f:\mathbb{R}^n\rightarrow{\mathbb{R}}$</span> is Lipschitz, an... | Cesareo | 397,348 | <p>Considering</p>
<p><span class="math-container">$$
f''+\alpha f'+\beta f = 0
$$</span></p>
<p>with <span class="math-container">$\alpha = \frac ba, \beta = \frac ca$</span> after solving the ODE with the established initial conditions we get</p>
<p><span class="math-container">$$
\frac{f_1}{f_t} = \frac{\left(\al... |
651,027 | <p><img src="https://i.stack.imgur.com/SIdAQ.jpg" alt="enter image description here"></p>
<p>I'm doing this Solving Abstract Problem but I'm not sure which one it is. I mean from the Series I can see there's a pattern but in the Options I don't see images that link with the Series. Do you have any idea - unless it's B... | lhf | 589 | <p>I think the answer is E.</p>
<p>At each step in the series, X goes round 45 degrees counterclockwise while the circle goes round 45 degrees clockwise. </p>
|
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | Adam Ledger | 504,652 | <p>If I could give a younger me advice, I would tell myself to lower my pride and accept as much help as I could in the times it was available to me, and shed my fear of humiliation and inferiority that kept me from even considering talking to others entirely let alone other mathematicians.</p>
<p>While you should str... |
331,543 | <blockquote>
<p>Given a fair six-sided die. Find the probability generating functions for the number of the throw on which the rth six appears. Hence find the probability that the fifth six occurs on the 20th throw. </p>
</blockquote>
<p>For the rth six is it true that $X~Geo((1/6)^r)$, so PGF is $(1/6)^rt+(1/6)^r\t... | Marko Riedel | 44,883 | <p>We will calculate the PGF as asked for in the problem. The probability that the $r$-th six occurs after $n$ rolls of the die is
$$ \frac{1}{6} \binom{n-1}{r-1}
\left(\frac{5}{6}\right)^{n-r} \left(\frac{1}{6}\right)^{r-1}.$$
Therefore the PGF is
$$p(z) = \sum_{n\ge r} \binom{n-1}{r-1}
\left(\frac{5}{6}\right)^{n... |
979,947 | <p>If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?</p>
<p>UPDATE: I've done some more work and have come up with the following: Based on the equations I combined to arr... | Pieter21 | 170,149 | <p>Always take the educated guess. You often can exclude one or two silly answers, and with chance 1 out of 3 you beat leaving it blank with a few points.</p>
<p>For example: what is the main chemical element on the moon?</p>
<ol>
<li>Water</li>
<li>Cheese</li>
<li>Oxygen</li>
<li>Iron</li>
<li>Silicon</li>
</ol>
<p... |
979,947 | <p>If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?</p>
<p>UPDATE: I've done some more work and have come up with the following: Based on the equations I combined to arr... | David | 175,745 | <p>Yes, if you have bad luck guessing then it would be better in that case to leave it blank. Since it is a SAT and not a SGT, they should penalize more for a wrong answer such as -1/2 so that if someone doesn't know an answer, they are less inclined to guess. Then they might get a better assessment of aptitude rathe... |
4,380,862 | <p>How can I solve the tangent point and <span class="math-container">$a$</span> when <span class="math-container">$f(x)=\left(\log_{a}{x}\right)^2$</span> is tangent to <span class="math-container">$g(x)=-ax+2$</span>?</p>
<p>Although this can be solved by substituting <span class="math-container">$a=e^2$</span> and <... | Dstarred | 955,900 | <p>One cannot continuously attack this question algebraically (there is no general solution for <span class="math-container">$x$</span> without <span class="math-container">$a$</span>), seeing that <span class="math-container">$(\log_a{x})^2 = -ax+2$</span>, will eventually end up in a form similar to<span class="math-... |
549,411 | <p>How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for</p>
<p>$$
\frac{d}{dx} \prod_{j = 1}^n \psi_j(x)
$$</p>
| Marv Turner | 104,945 | <p>Their intersection is the empty set</p>
|
245,789 | <p>Let $f$ be a function defined on the interval $(−1, 1)$ such that for all $x, y \in (−1, 1)$, $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ Suppose that $f$ is differentiable at $x = 0$, show that $f$ is differentiable on $(−1, 1)$.</p>
<p>Need your kind guidance and help. Thanks in advance!</p>
| Hagen von Eitzen | 39,174 | <p>Letting $x=y=0$, we obtain $f(0)=\frac{2f(0)}{1-f(0)^2}$, hence $f(0)=0$ or $1-f(0)^2=2$. Since the latter is impossible, $f(0)=0$.
Then we are given that the limit
$$\tag1\lim_{h\to0}\frac{f(h)}h=\lim_{h\to0}\frac{f(h)-f(0)}h=f'(0)$$
exists and of course even more so we have
$$\tag2\lim_{h\to0}f(h)=0.$$
We find
$$\... |
1,248,517 | <p>I try to solve the following problem: <br>
Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of $A_i$. <br>
Since the matrices are very large, computing the eigenvalues from scratch isn't efficient (and since the... | Adhvaitha | 228,265 | <p>We have
$$3n^2+7n+4 = (n+2)(3n+1) + 2$$
Hence, if $d$ is a common divisor of $3n^2+7n+4$ and $n+2$, then $d$ must divide $2$. Hence, $\gcd(3n^2+7n+4,n+2) = 1 \text{ or }2$. $3n^2+7n+4$ is always even. Further, $n+2$ has the same parity as $n$.</p>
<p>Hence, if $n$ is odd, $$\gcd(3n^2+7n+4,n+2) = 1$$
while if $n$ is... |
1,248,517 | <p>I try to solve the following problem: <br>
Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of $A_i$. <br>
Since the matrices are very large, computing the eigenvalues from scratch isn't efficient (and since the... | Mark Fischler | 150,362 | <p>Apply Euclid's algorithm:
$$3n^2 + 7n + 4 = (3n+1)(n+2) + 2$$
$$ \text{gcd }(3n^2 + 7n + 4) = \text{ gcd }(n+2,2)
$$
So the answer is $1$ if $n$ is odd, but $2$ if $n$ is even. </p>
|
3,584,927 | <p>If I have two isomorphic groups, can I write <span class="math-container">$A \xrightarrow{\sim} B$</span> rather than <span class="math-container">$A \cong B$</span> to mean "A is isomorphic to B", or is the arrow notation only used if I have a map <span class="math-container">$\varphi : A \xrightarrow{\sim} B$</spa... | PrincessEev | 597,568 | <p>If <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are isomorphic, then it implies the existence of an isomorphism from one to the other. Though, that said, I wouldn't use that notation since it seems to imply you're talking about an isomorphism rather than the fact they're isomor... |
1,501,717 | <p>Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that</p>
<blockquote>
<p>$$\sog(\sog(x)) = \log(x)$$
for all real $x>0$.</p>
</blockquote>
<p>My questions: </p>
<ul>
<li>Does such a function exist?</li>
<li>How do I compute it?</li>
</ul>
| Will Jagy | 10,400 | <p>EDIT: I got my copy of Hellmuth Kneser (1950) from the Göttingen digital repository, click to download just the article itself (rather than the whole journal issue) <a href="http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002175851" rel="nofollow noreferrer">LINK HELLMUTH KNESER 1950 Reele analytische... |
1,501,717 | <p>Imagine a real-valued semilog function $\DeclareMathOperator{\sog}{sog}\sog$ with the property that</p>
<blockquote>
<p>$$\sog(\sog(x)) = \log(x)$$
for all real $x>0$.</p>
</blockquote>
<p>My questions: </p>
<ul>
<li>Does such a function exist?</li>
<li>How do I compute it?</li>
</ul>
| Franklin Pezzuti Dyer | 438,055 | <p>This question already has a <em>great</em> answer, but I thought that you might like to see an actual piecewise construction of such a function (defined on $(-\ln(2),\infty)$). My function is not analytic, but it is continuous and differentiable on $\mathbb R$.</p>
<p>First notice that if we find a function $h$ wit... |
2,367,875 | <blockquote>
<p>Let $K$ be a field and $x$ be trnacendental over $K$. Compute
$[K(x):K(\frac{x^5}{1+x})]$.</p>
</blockquote>
<p>I've never came across questions like these. It's easy to see that this degree is at most $5$, since: $$x^5-\alpha(x+1)=0$$ ($\alpha$ being $\frac{x^5}{1+x}$). But how can I show the othe... | Ennar | 122,131 | <p>What you have to do is show that $f(X)=X^5-\alpha(X+1)$ is irreducible over $K(\alpha)$ and you are done.</p>
<p>This is because we would have $[K(\alpha)(x):K(x)][K(x):K(\alpha)] = [K(\alpha)(x):K(\alpha)] = 5$ and $K(x)\neq K(\alpha)$ would imply $[K(x):K(\alpha)]=5$ by $5$ being prime.</p>
<p>Now, $f$ is primit... |
2,367,875 | <blockquote>
<p>Let $K$ be a field and $x$ be trnacendental over $K$. Compute
$[K(x):K(\frac{x^5}{1+x})]$.</p>
</blockquote>
<p>I've never came across questions like these. It's easy to see that this degree is at most $5$, since: $$x^5-\alpha(x+1)=0$$ ($\alpha$ being $\frac{x^5}{1+x}$). But how can I show the othe... | pisco | 257,943 | <p>We only need to show that the polynomial $x^5-\alpha x - \alpha$ is irreducible over $K(\alpha)[x]$. Note that the polynomial is primitive, thus it suffices to show it is irreducible over $K[\alpha][x]$.</p>
<p>We use the idea of Eisenstein criterion, let $\varphi:K[\alpha]\to K$ be the evaluation map sending $\alp... |
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | Dong-gyu Kim | 561,790 | <p>How about using an induction on <span class="math-container">$n$</span>?</p>
<p>From the assumption, <span class="math-container">$a_1\leq a_2\leq \cdots \leq a_n$</span>, we derive some facts: (you mentioned)</p>
<ul>
<li><p><span class="math-container">$a_1=a_2 = 1$</span></p></li>
<li><p><span class="math-conta... |
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | CopyPasteIt | 432,081 | <p>The motivation for this work was thinking about Calvin Lin's bonus problems. </p>
<hr>
<p>Proposition 1: Let <span class="math-container">$(a_i)_{1\le i\le n}$</span> be a family of positive integers such that </p>
<p><span class="math-container">$$\tag 1 \sum_{i=1}^n a_i = 2n - k \; \text{ with } \,1 \le k \lt n... |
2,566,546 | <p>Show that for any $x_1 < x_2$ and $y_1 < y_2$ one has
$P(x_1 < X ≤ x_2, y_1 < Y ≤ y_2) = F(x_2, y_2) + F(x_1, y_1) − F(x_1, y_2) − F(x_2, y_1)$.</p>
<p>Would I just need to split the LHS to something that gives me the right?</p>
| Elias Costa | 19,266 | <p>If $P^2=P$ then $P\in\mathbb{R}^{n\times n}$ is root of equation
$$
P^2-P=0.
$$
But we have a bit more. If $P^2=P$ then
$$
\cdots=P^{2\cdot n}=\cdots=P^{4}=P^{2}=P^{}
$$
Then for any fixed $n$ and $n$ real numbers $a_1,a_2,\ldots, a_n$ such that $ a_1+a_2+\ldots+a_n=1$ we have that $P$ is root of equation
$$
a_n\... |
2,912,401 | <p>I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy sequence. </p>
<p>I know that a sequence of real numbers is <em>not</em> Cauchy if there exists an $\epsilon>0$ such ... | José Carlos Santos | 446,262 | <p>Let $(a_n)_{n\in\mathbb N}$ be your sequence. Take $\varepsilon=\frac12$. Given $N\in\mathbb N$, take $n\geqslant N$ such that $a_n$ is of the form $\frac k{k+1}$ for some $k\in\mathbb N$ and let $m=n+1$. Then $a_m=\frac1{k+2}$. Therefore,$$\left\lvert a_m-a_n\right\rvert=\frac k{k+1}-\frac1{k+2}\geqslant\frac12=\va... |
1,429,563 | <blockquote>
<p>Let <span class="math-container">$\{F_n\}_{n\geq 0}$</span> be the Fibonacci sequence.</p>
<p>Prove that the number of primes <span class="math-container">$p$</span> so that <span class="math-container">$p\mid F_{p-1}$</span> is infinite.</p>
</blockquote>
<hr />
<p>I tried to use induction, to no avail... | Jack D'Aurizio | 44,121 | <p>Assume that $p$ is a prime for which $5$ is a quadratic residue.</p>
<p>That is equivalent, by quadratic reciprocity, to $p\equiv\pm 1\pmod{10}$.</p>
<p>Since the explicit formula for Fibonacci numbers gives:
$$ F_n = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\r... |
1,696,713 | <p>I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. </p>
<p>$e^{-2\ln{\sin{x}}}$</p>
| Ittay Weiss | 30,953 | <p>If you are interested in causality, then the book "Causality: Models, Reasoning, and Inference" by Judea Pearl should be of interest to you. It's a little difficult to judge from your question what you are actually after, but my guess is you will find plenty of illumination in that book. </p>
|
209,760 | <p>Is it possible to transform this equation to give <span class="math-container">$R$</span>?
<span class="math-container">$$y=x\left[\frac{\left(1+\frac{R}{12}\right)^{12\times{25}}}{\frac{R}{12}}-1\right]$$</span> </p>
| Yoni Rozenshein | 36,650 | <p>Letting $w = \frac y x + 1$ and $r = \frac R {12}$, we are left with inverting
$$ w = \frac {(1+r)^{300}} r $$
$$ w = \frac {(1+r)^{300}} {(r^{1/300})^{300}} $$
$$ w^{1/300} = r^{-1/300} + r^{299/300} $$
Letting $z = r^{1/300}$ and multiplying by $z$ we have
$$ z^{300} - w^{1/300}z + 1 = 0$$</p>
<p>This is a trinom... |
3,672,814 | <p><span class="math-container">$g_n(y)=\sum_{k=1}^{n} \frac{\sin ky}{k},x \in [0,\pi]$</span>. So i try to find the pointwise limit function first, following this approach in this post <a href="https://math.stackexchange.com/questions/13490/proving-that-the-sequence-f-nx-sum-limits-k-1n-frac-sinkxk-is">Proving that th... | RRL | 148,510 | <p>You have shown correctly with the Dirichlet test that <span class="math-container">$(g_n)$</span> converges uniformly on the interval <span class="math-container">$[\delta, \pi]$</span> for any <span class="math-container">$\delta > 0$</span>. This relies on the fact that <span class="math-container">$\left|\sum... |
2,657,571 | <p>Apologies if this is a duplicate. I searched and didn't find anything quite like it.</p>
<p>Suppose I have a drawer with an equal number of N black socks and N white socks. They're all mixed up. So, my chances of picking a matching pair in the first two selections is (N-1)/(2N-1), right? Well, what if, <em>before</... | G Tony Jacobs | 92,129 | <p>It doesn't matter if the partitioning happens before or after the first draw. Suppose it happens after. Suppose also that the first sock drawn was black.</p>
<p>Now, we partition off $N$ from the remaining $2N-1$, to obtain our pool for the second draw. On average, the composition of this pool is $\frac{N-1}{2N-1}$... |
2,575,156 | <p>I thought of this question through an excercise in algebraic geometry where the rings were $\Gamma(V)$ and $\mathcal{O}_P(V)$, although my question is more general. If $R\subset S$ two (commutative with 1) rings (R subring of S) and I prime ideal in R, then consider I'=(I) the ideal in S, the ideal generated by I. I... | A Square | 497,575 | <p>As I don't have enough reputation for writing comments, I'll post it here.</p>
<p>It is not true in the general case. Take for example $\mathbb{Z}$ and the ideal generated by $(2)$ then this ideal in $\mathbb{Q}$ is just $\mathbb{Q}$.</p>
<p>You can find a plenty of examples in algebraic number theory.
For instan... |
153,999 | <p>How to solve: $|\sqrt{x-1}-2| + |\sqrt{x-1}-3|=1$.</p>
<p>I would like to know how to solve an absolute value equation when there is a square root sign inside.</p>
| Community | -1 | <p>Let us call $\sqrt{x-1}$ as $y$. Note that by definition $y \geq 0$. Now we need to find $y$ such that $$\lvert y-2 \rvert + \lvert y - 3 \rvert = 1$$
To solve this lets split into three cases.</p>
<ol>
<li>$y < 2$. This gives us that $$\lvert y-2 \rvert + \lvert y - 3 \rvert = (2-y) + (3-y) = 5 -2y > 5 -2 \t... |
1,675,411 | <p>So far I have this:</p>
<p>First consider $n = 5$. In this case $(5)^2 < 2^5$, or $25 < 32$. So the inequality holds for $n = 5$.</p>
<p>Next, suppose that $n^2 < 2^n$ and $n \geq 5$. Now I have to prove that $(n+1)^2 < 2^{(n+1)}$.</p>
<p>So I started with $(n+1)^2 = n^2 + 2n + 1$. Because $n^2 < 2... | Saikat | 230,452 | <p>First, we prove a simple lemma. </p>
<p>$$
\begin{align}
n & \gt 5\\
\implies n-1 &\gt 4\\
\implies (n-1)^2 &\gt 16 \gt 2 \\
\implies n^2 - 2n + 1 &\gt 2\\
\implies n^2 &\gt 2n +1\\
\end{align}
$$ </p>
<p>Now, we start with our induction step.
$$
\begin{align}
n^2 &\lt 2^n \text{ From... |
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | Arnaldo | 391,612 | <p><strong>Hint</strong></p>
<p>Write like this:</p>
<p>$$4^{k+2}-4^{k+1}+5^{2k+1}-5^{2k-1}=3\cdot4^{k+1}+24\cdot5^{2k-1}=3\cdot(4^{k+1}+5^{2k-1})+21.5^{2k-1}$$</p>
<p>Can you finish? </p>
<p>P.S: Use hyp induction for $k$ and note that $7|21$.</p>
<p>P.S: You can also prove that without use induction.</p>
|
2,736,596 | <p>I want to proof this equation below.
$$
f^{-1}(\bigcup B_{j})=\bigcup(f^{-1}B_{j})
$$
I know it's true but I don't have idea how to proof and explain.
Please help.</p>
| bsbb4 | 337,971 | <p>Let $x \in f^{-1}(\bigcup B_j)$. This means that there exists $y \in \bigcup B_j$ such that $f(x)=y$. For some $i$ in the index set of $\bigcup$, $y \in B_i \subset \bigcup B_j$. This gives that $ x \in f^{-1}(B_j)$, so we also get $x \in \bigcup f^{-1}(B_j)$.</p>
<p>Conversely, assume $x \in \bigcup f^{-1}(B_j)$. ... |
4,264,808 | <p>So we know that <span class="math-container">$\frac{n^2}{2} \geq \frac{n}{2}$</span>, but I'm stuck proving that <span class="math-container">$\frac{n^2}{2} \geq \frac{n^2}{2^{\sqrt{\log n}}}\geq \frac{n}{2}$</span>. Am I missing something?</p>
| lone student | 460,967 | <p>Let <span class="math-container">$\sqrt {\log n}=u,\thinspace u≥0$</span>, then we have</p>
<p><span class="math-container">$$\begin{align}&\frac{n^2}{2^{\sqrt{\log(n)}}} ≥ \frac{n}{2}\\
\iff &\frac{n^2}{2^{\sqrt{\log(n)}}} ≥ \frac{n^2}{2n}\\
\iff &n≥2^{\sqrt{\log(n)}-1}\\
\iff &e^{u^2}≥2^{u-1}\\
\if... |
3,245,854 | <p>Hi I am trying to solve an integral problem that involves trig substitution. First I tried completing the square, which gave me <span class="math-container">$1/\sqrt{(x+3)^2+2^2}$</span>. I know I am supposed to use <span class="math-container">$x = \arctan(\theta)$</span>. Does that mean it should be: </p>
<p><spa... | José Carlos Santos | 446,262 | <p>Do <span class="math-container">$x=2\tan(\theta)-3$</span> and <span class="math-container">$\mathrm dx=2\sec^2(\theta)\,\mathrm d\theta$</span>. You will get<span class="math-container">$$\int\frac{2\sec^2(\theta)}{\sqrt{4\tan^2(\theta)+4}}\,\mathrm d\theta=\int\sec(\theta)\,\mathrm d\theta.$$</span>Now, you can us... |
2,661,004 | <p>I have the following given as an answer to a questions, but I don't understand how I get $1+ \log n$ in the second step. I think I am to use L'Hospital's, but deriving the logarithm is $1/(n\ln(b))$, so I don't know why I'm keeping $\log(n)$. I know it's probably something basic, but what am I missing about that s... | Henno Brandsma | 4,280 | <p>You just ignore the part of the ball that sticks out. By definition of the subspace metric we just use $B_{(1,y]}(7,r) = \{x \in (1,7] : |x - 7| < r\} = (7-r, 7]$ for $r<6$. So instead of all points closer then $r$ to $7$ in all of the reals we just take those inside the subspace in question. </p>
<p>This co... |
131,583 | <p>My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any <strong>algorithm</strong>) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, Hodg... | Liviu Nicolaescu | 20,302 | <p>To write the equations in invariant form you first need to define the various objects involved in invariant form. Sometime this is not so obvious and new insight is needed. Here are two examples that hopefully will illustrate ways in which things can get tricky.</p>
<p>1.The Euler-Lagrange equations are ... |
131,583 | <p>My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any <strong>algorithm</strong>) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, Hodg... | Bazin | 21,907 | <p>Let $\mathcal M$ be a smooth manifold. A linear PDE on $\mathcal M$ is a sum of terms
$$
X_1\dots X_N u,\quad\text{where the $X_j$ are smooth vector fields.}
$$
We may use the convention that if $N=0$, this is just the multliplication by a smooth function.
In other words a linear PDE on $\mathcal M$ is a (non-commut... |
551,471 | <p>In a proof I am reading, the author considers the situation where $G$ is a reductive algebraic group (variety) over the complex numbers $\mathbb C$ and $N\trianglelefteq G$ is a closed, normal subgroup of $G$ which is also finite. The author concludes that <em>"because $G$ is reductive"</em>, $N$ must be contained i... | Community | -1 | <p>I recall writing about this a while ago. This holds for connected algebraic groups over an algebraically closed field $k$, not necessarily reductive: Let $H$ be the normal subgroup. Let $y\in H$ and let $G\to G$ be given by $x\mapsto xyx^{-1}$. Since $H$ is normal, this sends $G$ into $H$. The image of a connected g... |
3,441,647 | <p>I was asked to find the domain of <span class="math-container">$$\arcsin[\frac{x^2+1}{2x}]$$</span>
My first step was <span class="math-container">$$-1\leq\frac{x^2+1}{2x} \leq1$$</span></p>
<p>What I don't understand is why I cannot cross multiply to get <span class="math-container">$$-2x\le {1+x}^{2} \le 2x$$</sp... | Axion004 | 258,202 | <p>In order to find the domain, you need to check every possible real value of <span class="math-container">$x$</span>. This includes values of <span class="math-container">$x$</span> where <span class="math-container">$x\ge 0$</span> and where <span class="math-container">$x<0$</span>. When you multiplied all three... |
2,077,053 | <p>Please check my proof</p>
<p>If $H\triangleleft G$ then $gHg^{-1}=H $ for every $g\in G $</p>
<p>If $K\triangleleft G$ then $gKg^{-1}=K$ for every $g\in G$</p>
<p>Then $H\cap K = gHg^{-1}gKg^{-1}=gHKg^{-1}$</p>
<p>Then $H\cap K\triangleleft G$</p>
| George Law | 141,584 | <p>The third line is wrong: you seem to be claiming that $H\cap K=HK$, which isn't true (e.g. try $G=\mathbb Z/6\mathbb Z$, $H=\langle2\rangle$, $K=\langle3\rangle$).</p>
<p>What you should be doing is proving that $g(H\cap K)g^{-1}=H\cap K$. Hint: Take $a\in H\cap K$; then $a\in H$ and $a\in K$, so what can you say a... |
2,782,338 | <blockquote>
<p>Let $X_1, X_2, \cdots$ be independent and identically distributed random variables with expectation $\mu$. Let $N$ be a positive integer-valued random variable such that $E[N] < \infty$ and such that $I_{N≥n}$ is independent of $X_n$ for all $n$. Prove that $$E\!\left[\sum_{i=1}^NX_i\right]=\mu\, E... | Stefan Hansen | 25,632 | <p>First of all, $I_{N\geq n}$ is the random variable defined by
$$
I_{N\geq n}(\omega)=I_{N(\omega)\geq n}=\begin{cases} 1\quad\text{if } N(\omega)\geq n\\0\quad \text{if } N(\omega)<n\end{cases}
$$
and is often called an indicator function (for the set $\{N\geq n\}$). Now, led by the assumption that $I_{N\geq n}$ ... |
76,474 | <p>Use Newton's method to apporzimate the indicated root of the equation correct to six decimal places. The negative root of $e^x = 4-x^2$</p>
<p>I do not know what a negative root is nor do I really know what I am supposed to do. I am guessing raise everything by loge.</p>
| Ross Millikan | 1,827 | <p>The negative root is the one where $x<0$. This equation has two roots, one near $x=-2$ and one near $x=1$. You could look at <a href="http://www.wolframalpha.com/input/?i=plot%204-x%5E2-exp%28x%29" rel="nofollow">this plot</a></p>
<p>You are solving $0=4-x^2-e^x$. For Newton's method, you need to take the der... |
2,254,694 | <p>I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:</p>
<p>Attempt 1:</p>
<p>Write out the terms of $b_n$:</p>
<p>$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a_{3}-\frac{a_{2}}{2}$$
$$b_3=a_{4}-\frac{a_{3}}{2}$$
$$\cdots$$
$$... | Lakshya Gupta | 439,351 | <p>Hint :
If $L_t n > \infty$ $\frac{A_{n+1}}{A_{n}}$ is less than $1$. Then the sequence $A_n$ must converge.</p>
|
2,765,570 | <p>Let $E = \{\frac{1}{n} | n \in \mathbb{N}\}$. Show that the function $f(x) = 1 \text{ if } x \in E \text{ and } f(x) = 0 \text{ if } x \notin E$ is integrable on $[0,1]$.</p>
<p>I know of the definition of integrable but I'm having a hard time applying it to this function.</p>
| orangeskid | 168,051 | <p>Let $N \ge 1$. Consider a division of $[0,1]$ with $\|\Delta\| = \delta$. Let $(\xi_i)$ a system of intermediate points for $\Delta$. Let us estimate $S(f,\Delta, \xi)$. Consider $S_1$ the sum corresponding to the intervals of $\Delta$ that intersect $[0,\frac{1}{N}$. We have $|S_1|\le 1/N + \delta$. Let $S_2$ be th... |
495,064 | <p>I'm currently tackling the following problems:</p>
<p>a) Let $a,b \in \mathbb{Z}$ and let $m$ be a nonnegative integer. Prove that $(a,b)=1$ if and only if $(a^m,b)=1$.</p>
<p>If $(a^m,b)=1$ then $pa^m+qb=1$ for some integers $p$ and $q$. It follows that $(pa^{m-1})a+qb=1$, which shows that $(a,b)=1$. This took ca... | Pedro | 23,350 | <p>This really boils down to two important facts:</p>
<p>$(1)$ If $p$ is prime, then $p\mid ab\implies p\mid a $ or $p\mid b$.</p>
<p>$(2)$ We have $(a,b)>1$ iff there exists a prime $p$ such that $p\mid a $ and $p\mid b$ (, if and only if $p\mid (a,b))$.</p>
<p>Consequently, if $(a^m,b)>1, p\mid a^m\implies ... |
2,033,834 | <blockquote>
<p>Evaluate the work integral where $F(x,y)=\langle-y,x\rangle$ over a triangle with vertices $A(-2,-2)$, $B(2,-2)$, $C(0,1)$.</p>
</blockquote>
<p>I am not sure how to approach this problem.
I tried setting $AB(4,0)$, $BC(-2,3)$ and $CA(-2,-3)$ but I am not sure how to proceed.</p>
<p>Without using G... | John Wayland Bales | 246,513 | <p>The three segments $AB,BC,CA$ can each be parameterized on the interval $[0,1]$ as follows:</p>
<p>\begin{eqnarray}
(1-t)A+tB=(1-t)\langle-2,-2\rangle+t\langle2,-2\rangle&=&\langle-2+4t,-2\rangle\\
(1-t)B+tC=(1-t)\langle2,-2\rangle+t\langle0,1\rangle&=&\langle2-2t,-2+3t\rangle\\
(1-t)C+tA=(1-t)\lang... |
5,581 | <p>(Wade's rewording)</p>
<p>There is a phrase that contains $4$ words and $22$ total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and $12$ letters appear once each. What is this phrase?</p>
<blockquote>
<p>@@$$#... | Arturo Magidin | 742 | <p>As David Speyer notes, the problem is too unconstrained for a solution as it stands. You can make some guesses using frequency tables for letters in English, but that's about the only place where some math will come in.</p>
<p>If it were a "straight cryptogram" (the letters are placed in the correct order, whether ... |
2,963,560 | <blockquote>
<p>Given <span class="math-container">$a,b,c\in \Bbb Z$</span>, pairwise distinct, and <span class="math-container">$n\in \Bbb N\setminus\{0\}$</span> prove that
<span class="math-container">$$S(n)=\frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)}\in \Bbb Z.$$</span></p>
<p>Sou... | Batominovski | 72,152 | <p>Let <span class="math-container">$f(x)$</span> denote the remainder of <span class="math-container">$x^n$</span> when divided by <span class="math-container">$g(x):=(x-a)(x-b)(x-c)$</span>. Because <span class="math-container">$g(x)\in\mathbb{Z}[x]$</span> is monic and <span class="math-container">$x^n\in\mathbb{Z}... |
1,179,959 | <p>taking $v$ as $x$ axis and $u$ as $y$ axis I would like to know whether ${1\over v}+{1\over u}={1\over f}$ has graph of the form $xy=c^2$?</p>
| Rory Daulton | 161,807 | <p>Not quite.</p>
<p>The shape is indeed a hyperbola with horizontal and vertical asymptotes, as does $xy=c^2$, but the center is moved. The center of $xy=c^2$ is at the origin, while your function has the center at the point $(f,f)$. The vertices of your graph are at the origin and $(2f,2f)$.</p>
<p>We can show this... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Sangchul Lee | 9,340 | <p><strong>EDITED.</strong> Some simplifications were made.</p>
<hr>
<p>Here is a solution.</p>
<p><strong>1. Basic facts on the dilogarithm.</strong> Let $\mathrm{Li}_{2}(z)$ be the <em>dilogarithm</em> function defined by</p>
<p>$$ \operatorname{Li}_{2}(z) = \sum_{n=1}^{\infty} \frac{z^{n}}{n^{2}} = - \int_{0}^{z... |
167,228 | <p>I have shown several plots by Show function:</p>
<pre><code>Show[p1, p2, p3, p4, p5, p6, p7, p8, PlotRange -> All]
</code></pre>
<p>I need to save data of all plots in one <code>txt</code> format, so that they can be recognizable from each other. How can I do that?</p>
<p>Each plot is a 2D plot say, $f_i$ as a... | Akku14 | 34,287 | <p><strong>Improved answer:</strong></p>
<p>Find all roots and add conjugate complex roots to get all real factorisations.</p>
<pre><code>sr = Select[Root[x^5 + 1, #] & /@ Range[5], Im[#] == 0 &]
(* {-1} *)
si = Select[Root[x^5 + 1, #] & /@ Range[5], Im[#] != 0 &]
(* {Root[1 - #1 + #1^2 - #1^... |
4,531,507 | <p><a href="https://i.stack.imgur.com/rJmjH.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rJmjH.jpg" alt="enter image description here" /></a></p>
<blockquote>
<p>Find the area of <span class="math-container">$ABCD$</span> if <span class="math-container">$PSC$</span> is a semicircle.</p>
</blockqu... | timon92 | 210,525 | <p>Let <span class="math-container">$E$</span> be the projection of <span class="math-container">$D$</span> on <span class="math-container">$CQ$</span>. Move triangle <span class="math-container">$BCQ$</span> by <span class="math-container">$\vec{BA}$</span>. The question boils down to finding the area of a parallelogr... |
2,379,466 | <p>Let there be a set A ,where A = {0,1,2,3,4,5,6,7,8,9} and we are to choose a two member subset {k,m} from A such that k,m are elements of A</p>
<p>What is the probability that |k-m| = 2 ?</p>
| Graham Kemp | 135,106 | <p>Noting that the members in set $A$ are all of the integers between 0 and 9 inclusive</p>
<p>Thus we are being asked for the probability of selecting two members in the list that are seperated by one other member.</p>
<p>So how many ways are there to do this? How many equally-likely ways are there to select ... |
2,379,466 | <p>Let there be a set A ,where A = {0,1,2,3,4,5,6,7,8,9} and we are to choose a two member subset {k,m} from A such that k,m are elements of A</p>
<p>What is the probability that |k-m| = 2 ?</p>
| farruhota | 425,072 | <p>$1)$ All possible two member subsets: combination of $10$ digits taken $2$ at a time.</p>
<p>$2)$ All possible two member subsets satisfying $|k-m|=2: \{0,2\}, \{1,3\}, \cdots, \{7,9\}.$</p>
<p>$3)$ The probability is to divide $2)$ by $1)$ to get $8/45$.</p>
<p>Note: $\{0,2\}=\{2,0\}$. If insisted on distinct $k... |
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