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 |
|---|---|---|---|---|
759,810 | <p>I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?</p>
| lab bhattacharjee | 33,337 | <p>Finding the last two digits of $a$ essentially $\displaystyle a\pmod{100}$ </p>
<p>Now as $\displaystyle2012\equiv12, 2012^{2012}\equiv12^{2012}\pmod{100}$</p>
<p>Again as $\displaystyle(12,100)=4$ let us find $\displaystyle12^{2012-1}\pmod{\frac{100}4}$ i.e., $\displaystyle12^{2011}\pmod{25}$</p>
<p>Now using <... |
2,202,339 | <p>It's easy to prove $x^2+1$ is never divisible by $4k+3$ primes. I know a non-constructive proof for existing $x$ so that $p|x^2+1$ for $4k+1$ primes. is there any constructive one?</p>
| MR_BD | 195,683 | <p>$$ \: [(\frac{p-1}2)!]^2 \equiv -1 \:(mod \: p)$$</p>
<p>By Wilson's Theorem.</p>
|
2,005,604 | <p>Showing $\sqrt a + $$\sqrt {\cos(\sin a)} = 2$</p>
<p>I've attempted various manipulations (multiplying by one, squaring, etc.) but cannot find a way to solve for a. Anyone have an idea how I can approach this problem? Thanks. </p>
| Claude Leibovici | 82,404 | <p>For this kind of equation, there is no analytical solution (it is already the case for $x=\cos(x)$) and numerical methods should be used.</p>
<p>As Robert Israel already answered, Newton method could be the simplest to use considering $$f(x)=\sqrt{x}+\sqrt{\cos (\sin (x))}-2$$ Starting from a "reasonable" guess $x_... |
3,020,365 | <p>Let <span class="math-container">$A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$</span>.</p>
<p>Then </p>
<ol>
<li><p><span class="math-container">$\sup (A)<\frac{2}{\pi}+\frac{1}{n\pi}$</span> for all <span class="math-container">$n\ge 1$</span>.</p></li>
<li><p><span class="math-container">$\inf (A)>... | Calvin Khor | 80,734 | <p>The graph of <span class="math-container">$\sin(1/t)$</span> turns at <span class="math-container">$t=2/3\pi$</span>, but this is not true of <span class="math-container">$t\sin(1/t)$</span>. Multiplying by <span class="math-container">$t$</span> has the effect of pushing the minimum point to the right and down slig... |
3,020,365 | <p>Let <span class="math-container">$A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$</span>.</p>
<p>Then </p>
<ol>
<li><p><span class="math-container">$\sup (A)<\frac{2}{\pi}+\frac{1}{n\pi}$</span> for all <span class="math-container">$n\ge 1$</span>.</p></li>
<li><p><span class="math-container">$\inf (A)>... | zhw. | 228,045 | <p>Like some other answers, but written differently:</p>
<p>Let <span class="math-container">$f(t)= t\sin(1/t).$</span> Then <span class="math-container">$f'(t)= \sin(1/t) - [\cos (1/t)]/t$</span> for <span class="math-container">$t>0.$</span> Thus <span class="math-container">$f'(2/(3\pi)) = -1.$</span> Hence for ... |
16,749 | <p>I wanted to remove the <code>Ticks</code> in my coding but i can't. Here when i try to remove the <code>Ticks</code> the number also gone. I need numbers without <code>Ticks</code>, <code>Ticks</code> and <code>GridLines</code> should be automatic and don't use<code>PlotRange</code> .</p>
<pre><code>BarChart[{{1,... | Ronny | 521 | <p>You could specify the <code>Ticks</code> manually, for example by</p>
<pre><code>Ticks -> {Table[{i, i, 0}, {i, 0, 14, 2}], None},
</code></pre>
<p>The y axis stays without Ticks by the <code>None</code> the x axis gets at $i=0,2,...,14$ a Tick at $i$ with $i$ as label and zero ticks width.</p>
|
1,796,156 | <p>Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number<a href="http://mathworld.wolfram.com/FibonacciNumber.html" rel="noreferrer">$^{[1]}$</a><a href="http://en.wikipedia.org/wiki/Fibonacci_number" rel="noreferrer">$\!^{[2]}$</a><a href="http://oeis.org/A000045" rel="noreferrer">$\!^{[3]}$</a>. The Fibonacci number... | Zach466920 | 219,489 | <p>(<strong>Main Result</strong>: I have a formula for what you want, its practical and it works!)</p>
<p>Since you ask multiple questions, I suppose what follows isn't a complete answer. I learned this trick the other day, so I'm more than happy to share it here,</p>
<p>If you want an integral that can generalize th... |
269,548 | <p>I want to know how I can solve this or plot a versus b ?</p>
<pre><code>Solve[ Sqrt[a] Cosh[1.2 Log[1.65 Sqrt[1/a]]] == Sqrt[-b] Sinh[1.2 Log[1.65 Sqrt[-(1/b)]]], b]
</code></pre>
<p>Thanks</p>
| MarcoB | 27,951 | <p>Your current code mentions that <code>Solve</code> tried to rationalize your floating-point constants. If you try to do that by hand, then Solve will return <code>Solve::nsmet: This system cannot be solved with the methods available to Solve.</code></p>
<p>However, you can use <code>ContourPlot</code> to obtain a p... |
591,765 | <blockquote>
<p>What is the way to convince myself that $\left\langle(1,2),\ (1,2,3,4)\right\rangle=S_4$ but $\left\langle(1,3),\ (1,2,3,4)\right\rangle\ne S_4$?</p>
</blockquote>
<p>Let $\sigma$ be any transposition and $\tau$ be any $p-$cycle, where $p$ is a prime.
Then show that $S_p=\langle\sigma,\tau\rangle$.</... | Mikasa | 8,581 | <p>Besides to @Betty's points, there is another way for seeing why does this happen. We know that $S_4$ can have the following presentation:</p>
<p>$$S_4=\langle a,b\mid a^2=b^4=(ab)^3=1\rangle$$ Let's satisfy $a=(1,2),~~b=(1,2,3,4)$ in above relations. Indeed $a$ and $b$ can do that, but what will happen if we set $a... |
6,355 | <p>My question is located in trying to follow the argument bellow. </p>
<p>Given a normal algebraic variety $X$, and a line bundle $\mathcal{L}\rightarrow X$ which is ample, then eventually such a line bundle will have enough section to define an embedding $\phi:X\rightarrow \mathbb(H^0(X,\mathcal{L}^{\otimes d}))=\ma... | Sándor Kovács | 10,076 | <p>Let $R(X,\mathcal L)=\oplus_d H^0(X,\mathcal L^{\otimes d})$ as graded rings, $s_0,\dots s_m\in H^0(X,\mathcal L^{\otimes d})$ and finally $R(X,\mathcal L,s_{\cdot})$ the (graded) subring generated by the $s_0,\dots s_m$ in $R(X,\mathcal L)$. </p>
<p>Facts:</p>
<ol>
<li><p>${s_0,\dots,s_m}$ define a rational map ... |
4,413,641 | <p>I'm trying to understand a proof that given a vector space <span class="math-container">$V$</span> over the field <span class="math-container">$F$</span> and <span class="math-container">$n$</span> vectors <span class="math-container">$v_1, \ldots, v_n$</span>, <span class="math-container">$\mathrm{span}(v_1, \ldots... | Dr. Sundar | 1,040,807 | <p>Here, <span class="math-container">$v_1, v_2, \ldots, v_n$</span> are <span class="math-container">$n$</span> vectors.</p>
<p>Let us fix some notation first.</p>
<p>We define <span class="math-container">$T = \mbox{span}\left\{ v_1, v_2, \ldots, v_n \right\}$</span>.</p>
<p>It is easy to verify that <span class="mat... |
3,056,121 | <p>I'm trying to find a function with infinitely many local minimum points where x <span class="math-container">$\in$</span> [0,1] and f has only 1 root. No interval should exist where the function is constant.</p>
| Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
705,945 | <p>I have this expression:
$$\sum_{\{\vec{S}\}}\prod_{i=1}^{N}e^{\beta HS_{i}}=\prod_{i=1}^{N}\sum_{S_{i}\in\{-1,1\}}e^{\beta HS_{i}} \qquad (1)$$
Where $\sum_{\{\vec{S}\}}$ means a sum over all possible vectors $\vec{S}=(S_1,...,S_N)$ with the restriction that $S_i$ can only take the values $\{-1,+1\}$, i.e. the sum i... | Ana S. H. | 48,197 | <p>The problem was in how to write down the sum $\sum_{\{\vec{S}\}}$. Since this sum is over all the possible vectors $\vec{S}=(S_1,...,S_N)$, (where $S_i\in\{-1,1\}$), we can rewrite this sum like</p>
<p>$$\sum_{S_{1}\in\{-1,1\}}\cdots\sum_{S_{N}\in\{-1,1\}}$$
i.e.
$$\sum_{\{\vec{S}\}}\prod_{i=1}^{N}e^{\beta HS_{i}}=... |
4,330,031 | <p>I want to prove that this degree sequence <span class="math-container">$(5,5,5,2,2,2,1)$</span> isn't valid to draw a graph from it, the graph needs to be simple. I am looking for a Theroem or a way to contradict the assumption that we can make a graph from it.</p>
<p>My solution was the following, for the given nod... | markvs | 454,915 | <p>There are matrices <span class="math-container">$A$</span> such that <span class="math-container">$A=A^{-1}$</span>. More generally if <span class="math-container">$\det A=\pm 1$</span> then <span class="math-container">$\det(ABA)=\det B$</span> for every <span class="math-container">$B$</span> (in fact the last... |
2,863,533 | <blockquote>
<p>Let $f(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients and all real roots, also $|f(i)|=1$ where $i=\sqrt{-1}$. Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$.</p>
</blockquote>
<hr>
<p>As $f(i)=-i-a+ib+c=1$ and $f(i)=-i-a+ib+c=-1$<br><br>
I don't know how ... | farruhota | 425,072 | <p>Alternatively, note that for $z=a+bi\in \mathbb Z, \bar{z}=a-bi \in \mathbb Z$, the norm is:
$$|z|=\sqrt{z\cdot \bar{z}}=\sqrt{a^2+b^2}.$$
So:
$$|f(i)|=|c-a+(b-1)i|=\sqrt{(c-a)^2+(b-1)^2}=1 \Rightarrow (c-a)^2+b^2-2b=0 \tag{1}$$
Let $x_1,x_2,x_3$ be the roots of $f(x)=0 \iff x^3+ax^2+bx+c=0.$
By the Vieta's formula... |
433,639 | <p>(What follows is motivated by an answer to <a href="https://mathoverflow.net/questions/433612/fourier-optimization-problem-related-to-the-prime-number-theorem?noredirect=1#comment1116702_433612">Fourier optimization problem related to the Prime Number Theorem</a>)</p>
<p>Let <span class="math-container">$f:\mathbb{R... | Alexandre Eremenko | 25,510 | <p>One can prove that under your assumptions
<span class="math-container">$$A:=\int_{-\infty}^\infty|x|f(x)dx\geq 9/(4\pi),$$</span>
but estimate is not exact.</p>
<p>The proof is based on the formula
<span class="math-container">$$A=-\lim_{y\to 0+}\frac{1}{\pi}\frac{d}{dy}\left(y\int_{-1}^1\frac{\hat{f}(t)}{t^2+y^2}dt... |
4,515 | <p>I've been using the sentence:</p>
<blockquote>
<p>If a series converges then the limit of the sequence is zero</p>
</blockquote>
<p>as a criterion to prove that a series diverges (when $\lim \neq 0$) and I can understand the rationale behind it, but I can't find a <strong>formal proof</strong>.</p>
<p>Can you h... | JT_NL | 1,120 | <p>Yes.</p>
<p>$$\lim_{n \to \infty} \left ( \sum_{k = 1}^{n + 1} a_k - \sum_{k = 1}^{n} a_k \right ) = \lim_{n \to \infty} a_{n + 1} $$
And both sums will converge to the same number so the limit is zero. This is by far the easiest proof I know.</p>
<p>This is the Cauchy criterion in disguise by the way, so you coul... |
463,239 | <p>Integrate $$\int{x^2(8x^3+27)^{2/3}}dx$$</p>
<p>I'm just wondering, what should I make $u$ equal to?</p>
<p>I tried to make $u=8x^3$, but it's not working. </p>
<p>Can I see a detailed answer?</p>
| amWhy | 9,003 | <blockquote>
<p>$$\int{x^2(8x^3+27)^{2/3}}dx$$</p>
</blockquote>
<p>It is certainly possible to work with $u = 8x^3$, but I'd suggest setting $u = 8x^3 + 27$. The key is to remember to compute and account for $\,du$. What you'll see is that for both $u = 8x^3$ and $u = 8x^3 + 27$, we have $du = 24x^2$. </p>
<p>$$... |
3,681,916 | <p>Given a matrix A, Does
<span class="math-container">$$\lim_{n \to \infty}A^n=0$$</span>
Imply that(by lower then I mean that every number in the lower matrix is closer or the same distance to 0 then it’s counterpart in the bigger one)
<span class="math-container">$$\text{if }a>b>0 \text{ then }A^a<A^b$$</sp... | Community | -1 | <p>Consider <span class="math-container">$A=e^{u}\begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix}$</span>, which we'll call <span class="math-container">$A(u,\alpha)$</span>. Such matrices have the property that <span class="math-container">$(A(u,\alpha))^n=A(nu,n\alpha)$</span> and <s... |
4,622,956 | <p>I think <span class="math-container">$\,9\!\cdot\!10^n+4\,$</span> can be a perfect square, since it is <span class="math-container">$0 \pmod 4$</span> (a quadratic residue modulo <span class="math-container">$4$</span>), and <span class="math-container">$1 \pmod 3$</span> (also a quadratic residue modulo <span cla... | Sai Mehta | 1,137,054 | <p>Assume that <span class="math-container">$9\cdot10^n+4\equiv4$</span> is a perfect square.<br />
<span class="math-container">$9\cdot10^n+4\equiv4\pmod9$</span>, so <span class="math-container">$9\cdot10^n+4$</span> can be represented as <span class="math-container">$(9m-2)^2=81m^2-18m+4$</span> or <span class="math... |
1,038,076 | <p>Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$?</p>
<p>I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, but how can I make use of this? Thank you so much.</p>
| Rajkumar | 184,827 | <p>Plotting $7 \times 13 \times 19 = a^2 − ab + b^2$, You will get an ellipse like this: </p>
<p><img src="https://i.stack.imgur.com/H5m6B.gif" alt="enter image description here"></p>
<p>.<br>
But if we apply the condition $a$ & $b > 0$, both will be positive in only first Quadrant. And again $a > b$, we... |
3,060,456 | <p>Can any one explain me the intuition behind this formula ? (with permutation example)</p>
<pre><code>P(n, k) = P(n-1, k) + k* P(n-1, k-1)
</code></pre>
| Wuestenfux | 417,848 | <p>Okay, define a word <span class="math-container">$x=x_1\ldots x_k$</span> of length <span class="math-container">$k$</span> over the alphabet <span class="math-container">$[n]=\{1,\ldots,n\}$</span> to be a <span class="math-container">$k$</span>-permutation of <span class="math-container">$n$</span> if <span class=... |
4,163,003 | <p>Let <span class="math-container">$Z \in \mathbb{R}^2$</span> be an i.i.d. Gaussian vector with mean <span class="math-container">$M$</span> where <span class="math-container">$P_{Z\mid M}$</span> is its distribution.</p>
<p>Let <span class="math-container">$g: \mathbb{R}^2 \to \mathbb{R}$</span> and consider the fol... | Michael Hardy | 11,667 | <p>This will not answer the question as stated, but whoever asks such a question may be interested in what I say here.</p>
<p>First, a random variable with a "Gaussian" or "normal" distribution need not have variance <span class="math-container">$1,$</span> so I would have stated what the variance i... |
3,545,548 | <p><span class="math-container">$\def\LIM{\operatorname{LIM}}$</span>
Let <span class="math-container">$(X,d)$</span> be a metric space and given any cauchy sequence <span class="math-container">$(x_n)_{n=1}^{\infty}$</span> in <span class="math-container">$X$</span> we introduce the formal limit <span class="math-cont... | Arjun | 744,667 | <p>If <span class="math-container">$x,y,z$</span> are succively in AP then let each
of them be<span class="math-container">$a,a+d,a+2d$</span></p>
<p>Which means
<span class="math-container">$$3^x,3^y,3^z=3^a,3^{a+d},3^{a+2d}$$</span></p>
<p>Which when written simply is
<span class="math-container">$$3^a,3^d×3^a,3... |
2,904,603 | <p>I'm working on the following question:</p>
<blockquote>
<p>Show that $G$ is a group if and only if, for every $a, b \in G$,
the equations $xa = b$ and $ay = b$ have solutions $x, y \in G$.</p>
</blockquote>
<p>I'm having trouble getting started because I'm not understanding what it means for "the equations $xa... | Sarvesh Ravichandran Iyer | 316,409 | <p>"the equations $xa = b$ and $ay = b$ have solutions $x,y \in G$", means precisely this :</p>
<blockquote>
<p>For all $a \in G$ and $b \in G$, <em>there exist</em> $x \in G$ and $y \in G$ which satisfy the equations $xa = b$ and $ay = b$ respectively.</p>
</blockquote>
<p>Since it only says that there is <em>at l... |
3,292,918 | <p>Let <span class="math-container">$X$</span> be a Banach space, and denote by <span class="math-container">$B_r (x)$</span> the closed ball of radius <span class="math-container">$r > 0$</span>
around <span class="math-container">$x \in X$</span>. Furthermore, let <span class="math-container">$A \subset X$</span> ... | mathworker21 | 366,088 | <p>I don't feel too bad posting a "proof by references" since I worked on this problem for quite a while but failed, before realizing the <span class="math-container">$N=1$</span> case seems pretty google-able. In any event, it seems Konyagin proved that any non-reflexive Banach space admits an equivalent norm and some... |
1,608,299 | <p>I am completely stuck on a question. I've done it 4 times, each times got different result, but never correct.</p>
<p>The third term of an arithmetic progression is 71 and the seventh term is 55. Find the sum of the first 45 terms.</p>
<p>Any ideas? Thanks</p>
| Archis Welankar | 275,884 | <p>$a+2d=71...1 $ while $a+6d=55$ solving you get $d=-4,a=79$ thus now sum =$\frac{45}{2}(158+(44)(-4))=-405$ thus its solved</p>
|
1,608,299 | <p>I am completely stuck on a question. I've done it 4 times, each times got different result, but never correct.</p>
<p>The third term of an arithmetic progression is 71 and the seventh term is 55. Find the sum of the first 45 terms.</p>
<p>Any ideas? Thanks</p>
| Guillemus Callelus | 361,108 | <p>We know that $a_3=71$ and $a_7=55$. Then,
$$a_3=a_1+(3-1)d\Rightarrow 71=a_1+2d$$
$$a_7=a_1+(7-1)d\Rightarrow 55=a_1+6d$$
If we subtract the two equations:
$$71-55=(a_1+2d)-(a_1+6d)\Rightarrow 16=-4d\Rightarrow d=-\dfrac{16}{4}=-4$$
Then,
$$71=a_1+2d\Rightarrow a_1=71-2d=71-2(-4)=71+8=79$$
The general term of the ar... |
228,135 | <p>I'm working to understand the Grothendieck topology version of the Zariski topology of schemes. Explained simply, it replaces the notion of "open subschemes" with "open immersions". So instead of $U\subseteq X$, we have $U\hookrightarrow X$.</p>
<p>The intersection $U\cap V$ between two open subschemes is replaced ... | Community | -1 | <p>A covering of $U$ is replaced by a family of morphisms $U_i\to U$ the union of whose images is $U$. </p>
|
1,579,616 | <p>So I know it's true for $n = 5$ and assumed true for some $n = k$ where $k$ is an interger greater than or equal to $5$.</p>
<p>for $n = k + 1$ I get into a bit of a kerfuffle.</p>
<p>I get down to $(k+1)^2 + 1 < 2^k + 2^k$ or equivalently:</p>
<p>$(k + 1)^2 + 1 < 2^k * 2$.</p>
<p>A bit stuck at how to pro... | Jimmy R. | 128,037 | <p>$$2^{k+1}=2\left(2^k\right)>2(k^2+1)=2k^2+2=k^2+k^2+2\overset{(*)}>k^2+2k+2=(k^2+1)+1$$ So, it remains to show that $k^2>2k$ for $k>4$. But $$k^2-2k=k(k-2)>0$$ for all $k>2$.</p>
|
164,213 | <p>Suppose I have some list with duplicates by some condition and I want to take the duplicates and apply some function to choose which duplicate to keep. Is there an efficient way to apply this transformation?</p>
<p>To clarify here is an example. Consider a list with elements with duplicate first elements:</p>
<pre... | Carl Woll | 45,431 | <p>You could use <a href="http://reference.wolfram.com/language/ref/GroupBy" rel="noreferrer"><code>GroupBy</code></a>:</p>
<pre><code>GroupBy[list, First, First@*MaximalBy[Last]]
Values @ %
</code></pre>
<blockquote>
<p><|2 -> {2, 0.2}, 3 -> {3, 0.4}, 4 -> {4, 0.9}, 6 -> {6, 0.3}|></p>
<p>{{2, 0.2}, {3, 0.... |
1,056,038 | <blockquote>
<p>Each of $n$ balls is independently placed into one of $n$ boxes, with all boxes equally likely. What is the probability that exactly one box is empty? (Introduction to Probability, Blitzstein and Nwang, p.36).</p>
</blockquote>
<ul>
<li>The number of possible permutations with replacement is $n^n$</l... | d125q | 112,944 | <p>You can think of the number of favorable arrangements in the following way: choose the empty box in $\binom{n}{1}$ ways. For each such choice, choose the box that will have at least $2$ balls (there has to be one such box) in $\binom{n - 1}{1}$ ways. And for this box, choose the balls that will go inside in $\binom{... |
4,615,947 | <p>Let <span class="math-container">$a,b\in\Bbb{N}^*$</span> such that <span class="math-container">$\gcd(a,b)=1$</span>. How to show that <span class="math-container">$\gcd(ab,a^2+b^2)=1$</span>?</p>
| giorgiokyn23 | 1,134,657 | <p>Complex Analysis approach: Let <span class="math-container">$$f(z)=\frac{e^{iz}}{z^2+2z+2}$$</span>Note that <span class="math-container">$Imm(f(z))=\frac{sin(z)}{z^2+2z+2}$</span>.
<span class="math-container">$f$</span> has 2 simple pole with order 1. The pole are <span class="math-container">$z_1=-1+i$</span> and... |
1,747,525 | <p>Given a number $N$, how can I write down a summation of all odd numbers divisible by 5 which are also less than $N$?</p>
<p>For instance, if $N = 27$ then I am looking for a series to generate $5+15+25$.</p>
<p>Its pretty clear the series looks like </p>
<p>$$\sum_{k=0}^{???} 5(2k+1)$$</p>
<p>but I am having tro... | sayan | 312,099 | <p>We have to find the sum of all odd numbers less than N and divisible by 5</p>
<p>where N is given. For a given N you can find the number of odd numbers divisible</p>
<p>by 5 is $$\left[\frac{N}{5}\right]-\left[\frac{N}{10}\right]$$ So the reqired sum is equal to $$\sum_{i=1}^{\left[\frac{N}{5}\right]-\left[\frac{N... |
505,178 | <blockquote>
<p>Suppose $k$ is an algebraically closed field, and $f\in k[x, y]$ is
an irreducible polynomial in two variables. Furthermore, suppose that $f(u(x),
v(y))=f(x, y)$ for every $x, y\in k$, where $u\in k[x]$, $v\in k[y]$ are
polynomials of one variable. Can we conclude that either $u(x)=x$ or
$v(y)=y... | 2'5 9'2 | 11,123 | <p>You can have $f(x,y)=x+y$, and then $u$ can shift by a constant, with $v$ shifting in reverse. </p>
|
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | Community | -1 | <p>You can take
<span class="math-container">$n-\lfloor n(\pi-3)\rfloor$</span> times <span class="math-container">$3$</span> and <span class="math-container">$\lfloor n(\pi-3)\rfloor$</span> times <span class="math-container">$4$</span> and you get an error smaller than <span class="math-container">$\dfrac1n$</span>, ... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | ati | 175,259 | <p>You shouldn't judge yourself or your achievements by other people's standards. Do mathematics because you enjoy it, and find it interesting. Many great discoveries have come from average minds in a spirit of idle curiosity and playfulness. Likewise many brilliant minds achieve very little. In research, your approach... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | reggaemuffin | 175,350 | <p>I was a bit like you, always excelling at math in school, taking extra courses etc. but I am exactly the opposite of you. I love solving challenges but fail doing things 'like in the handbook' and so I found computer science for myself. There I can solve problems in many different ways and challenge my mind while st... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | pocketlizard | 173,176 | <p>It doesn't really matter if you decide your career while in undergrad. You're there to develop a strong baseline and to learn about what you don't know yet. Honestly the one piece of advise I give to most undergrads is to take more math courses, even if it's unrelated to their field of study, because math will open ... |
2,207,848 | <p>I'm not very familiar with contraposition and so I am having some difficulties proving the statement. </p>
<blockquote>
<p>If $n$ is a positive integer such that $n \equiv 2 \pmod{4}$ or $n \equiv 3 \pmod{4}$, then $n$ is not a perfect square.</p>
</blockquote>
<p>What would be a good way to prove this?<br>
Need... | Bram28 | 256,001 | <p>In general, the contrapositive of a conditional:</p>
<p>'If P then Q' </p>
<p>is the statement:</p>
<p>'If not Q then not P'</p>
<p>Applied to your statement, we would thus get:</p>
<p>'If $n$ is a perfect square, then $n$ is not a positive integer such that $n \equiv 2 \pmod{4}$ or $n \equiv 3 \pmod{4}$'</p>
... |
3,383,206 | <p><strong>Question</strong>: Can <span class="math-container">$\int_0^\infty \frac{\sqrt{x}}{(1+x)^2} dx$</span> be computed with residue calculus?</p>
<p>The integral comes from computing <span class="math-container">$\mathbb{E}(\sqrt{X})$</span> where <span class="math-container">$X=U/(1-U)$</span> and <span class=... | paw88789 | 147,810 | <p>It's not enough to say there is a neighborhood of the point that intersects the set. You need <span class="math-container">$\underline {\rm{every}}$</span> neighborhood of the point to intersect the set.</p>
|
4,036,558 | <p><span class="math-container">$f(x)=e^x(x^2+x)$</span>, derive <span class="math-container">$\dfrac{d^n\,f(x)}{dx^n}$</span></p>
<p>may use Leibniz formula but i'm not sure:(</p>
| Mathematician 42 | 155,917 | <p>Just by calculating <span class="math-container">$f',f''$</span> and <span class="math-container">$f'''$</span> by hand, you can see the following pattern:
<span class="math-container">$$\frac{\mathrm{d}^nf}{\mathrm{d}x^n}(x) = e^x(x^2+(2n+1)x+n^2).$$</span></p>
<p>Now, we can prove this by induction (the base steps... |
4,036,558 | <p><span class="math-container">$f(x)=e^x(x^2+x)$</span>, derive <span class="math-container">$\dfrac{d^n\,f(x)}{dx^n}$</span></p>
<p>may use Leibniz formula but i'm not sure:(</p>
| zwim | 399,263 | <p>You can notice that by derivation we will get <span class="math-container">$e^xP(x)$</span> with <span class="math-container">$P$</span> polynomial, also the degree of <span class="math-container">$P$</span> stays unchanged since <span class="math-container">$(e^x)'$</span> does not bring some extra <span class="mat... |
2,227,280 | <p>For every positive number there exists a corresponding negative number. Would that imply that the number of positive numbers is "equal" to the number of negative numbers? (Are they incomparable because they both approach infinity?)</p>
| celtschk | 34,930 | <p>Yes, the existence of a one-to-one and onto mapping is exactly how equality of the size of sets (the technical term is "cardinality" is defined. The (cardinal) number of negative integers is the same as the cardinal number of positive integers, and the cardinal number of negative real numbers is the same as the card... |
1,590,625 | <blockquote>
<p>If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then
find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$.</p>
</blockquote>
<p><strong>My Attempt</strong>
$$f \left(\cfrac{3x+x^3}{1+3x^2}\right)=\log\left(\cfrac{1+\cfrac{3x+x^3}{1+3x^2}}{1-\cfrac{3x+x^3}{1+3x^2}}\rig... | SchrodingersCat | 278,967 | <p>$$\log(1+3x^2+3x+x^3)-\log(1+3x^2-3x-x^3)=\log(1+x)^3-\log(1-x)^3=3[\log(1+x)-\log(1-x)]=3f(x)$$</p>
|
69,378 | <p>Updated Question : How to show that in TH we never reach a state where there are no paths to the solution? ( without reversing moves, as if reversing is allowed this becomes trivial )</p>
<p>Edit : Thanks to <strong>Stéphane Gimenez</strong> for pointing out the distinction between “A deadlock would never occur” an... | davidlowryduda | 9,754 | <p>Well, one can give an exact algorithm to solve it. But I would recommend thinking of it as a recursive problem. Say we look at the 3 disk case. Well, that's really a 2 disk case on top of a third disk. So suppose we move those top 2 disks to another rod. Then we move the forgotted third to the desired end rod, and t... |
2,621 | <p>Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).</p>
<p>As a converse of the above, let ... | John D. Cook | 247 | <p>Monte Carlo methods are very useful in numerically evaluating high-dimensional integrals. With traditional integration methods, the number of integrand evaluations required to maintain accuracy grows quickly as dimension increases. With Monte Carlo integration, the number of integrand evaluations needed is <em>inde... |
700,012 | <p>Ok, so the question is to prove by induction that:</p>
<p>$${n \choose k} \le n^k$$</p>
<p>Where $N$ and $k$ are integers, $k \le n$;</p>
<p>How do I approach this? Do i choose a $n$ and a $k$ to form my base case?</p>
| RghtHndSd | 86,816 | <p>While there are various ways to approach this, I would recommend fixing (an arbitrary) $n$ and inducting on $k$. So your base case would then be to show that $\binom{n}{0} \leq n^0$. To do the inductive step, figure out what you would need to multiply $\binom{n}{k-1}$ by to get $\binom{n}{k}$.</p>
|
2,436,336 | <p>I am a bit puzzled. Trying to solve this system of equations: </p>
<p>\begin{align*}
-x + 2y + z=0\\
x+2y+3z=0\\
\end{align*}</p>
<p>The solution should be \begin{align*}
x=-z\\
y=-z\\
\end{align*} </p>
<p>I just don't get the same solution. Please advice.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>adding both equations we get
$$y=-z$$ plugging this in the first equation we get $$-x-2z+z=0$$ tgherefore $$x=-z$$</p>
|
1,344,284 | <p>if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my book such as $\left | Z-8 \right | +\left | Z+8 \right |=20$,$\left | Z-2 \right | = \left | Z-3i \right |$,$\left | 2Z... | user251257 | 251,257 | <p>It is true.
Let $R$ be the symmetric root of $A$, which exists positive for semidefinite matrices. That is, $R$ is symmetric, positive semidefinite, and $R^2=A$. </p>
<p>For $x\in\mathbb R^n$ we have
$$ x^T A x - x^T A e e^T A x \ge 0 $$
if and only if
$$ \|Rx\|^2 \underbrace{\|Re\|^2}_{\langle A,J \rangle=1} = x^... |
4,213,207 | <p>I am trying to find all homomorphisms from <span class="math-container">$\mathbb{Z}_{20}$</span> to <span class="math-container">$\mathbb{Z}_8$</span>. I understand how to do it - one completely determines any homomorphism, say <span class="math-container">$\phi$</span>, by computing multiples of <span class="math-c... | Community | -1 | <p>By the First Homomorphism Theorem, the quotient <span class="math-container">$\Bbb Z_{20}/\operatorname{ker}\phi$</span> is isomorphic to the image of <span class="math-container">$\phi$</span>. Since <span class="math-container">$|\operatorname{im}\phi|$</span> must be a common divisor of <span class="math-containe... |
2,894,126 | <blockquote>
<p>$$\int \sin^{-1}\sqrt{ \frac{x}{a+x}} dx$$</p>
</blockquote>
<p>We can substitute it as $x=a\tan^2 (\theta)$ . Then:</p>
<p>$$2a\int \theta \tan (\theta)\sec^2 (\theta) d\theta$$</p>
<p>Using integration by parts will be enough here. But I wanted to know if this particular problem can be solved by... | Community | -1 | <p>I don't see anything lengthy here.</p>
<p>$$\int \theta\frac{\sin\theta}{\cos^3\theta}d\theta=-\frac\theta{2\cos^2\theta}+\frac12\int\frac{d\theta}{\cos^2\theta}$$ and the last integral is immediate.</p>
|
1,729,308 | <p>The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is </p>
<p>$a)$ $3$</p>
<p>$b)$ $4$</p>
<p>$c)$ $5$</p>
<p>$d)$ $6$</p>
<p>My approach : I used the formula for the first $n$ terms of an A.P. ... | lulu | 252,071 | <p>To summarize the (extensive) discussion in the comments:</p>
<p>The OP's method is sound and nearly complete. To finish it off we look at the relation $$612=y^2-(a-1)^2=(y+(a-1))(y-(a+1)$$ To solve that (over the integers) we simply need to factor $612=cd$ where the factors must have the same parity. There are t... |
1,729,308 | <p>The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is </p>
<p>$a)$ $3$</p>
<p>$b)$ $4$</p>
<p>$c)$ $5$</p>
<p>$d)$ $6$</p>
<p>My approach : I used the formula for the first $n$ terms of an A.P. ... | Hypergeometricx | 168,053 | <p>Interesting question (+$1$). </p>
<p>As the common difference is $2$, the series is either one of odd numbers or even numbers only. As the sum is an odd number, it must be a series of odd numbers with an odd number of terms. This narrows it down to $3$ terms or $5$ terms from the choices given. </p>
<p>Since $153... |
4,013,065 | <p>I need help interpreting the answer to a question about the base and dimension of a subspace within linear algebra. I have a subspace W of <span class="math-container">$R^5$</span> that is spanned by the vectors:</p>
<p><span class="math-container">$${v_1}=\begin{pmatrix} 1 \\ 2 \\ 3 \\ -1 \\ 1 \end{pmatrix} , {v_2... | Timur Bakiev | 855,963 | <p>The dimension is exactly three since you have three linearly independent vectors <span class="math-container">$v_1, v_2, v_3$</span> in the generating system of <span class="math-container">$W$</span> and <span class="math-container">$v_4 = -v_2-v_3$</span>.</p>
|
20,982 | <p>Let E be an elliptic curve over a finite field k (char(k) is not 2) be given by y^2 = (x-a)(x-b)(x-c) where a,b and c are distinct and are in k. Then why is (c,0) is in [2]E(k) iff c-a and c-b is a square in k-{0}? </p>
| Bjorn Poonen | 2,757 | <p>Here is a conceptual explanation that applies to any $E/k$ with $\operatorname{char} k \ne 2$ and $E[2] \subseteq E(k)$, and that explains why $x-a$ and $x-b$ are the relevant rational functions (the field $k$ need not be finite).</p>
<p>The map $[2]\colon E \to E$ makes the function field $k(E)$ a finite extension... |
1,873,596 | <p>Near the end of <a href="http://www.maa.org/sites/default/files/pdf/upload_library/2/Rice-2013.pdf" rel="nofollow noreferrer">this MAA piece about elliptic curves</a>, the author explains why the complex domain of the cosine function is a sphere: since it's periodic, its domain can be taken as a cylinder, wrapping u... | Max | 2,633 | <p>Let me start by saying you are right in a sense, and I think the article is at best being unclear, but the larger point that the equation of ellipse cuts out a sphere is also right. There are a few things going on, hence a long answer below.</p>
<p>What one is trying to describe is the topology of a set defined by... |
166,460 | <p>I work with linear combinations of graphs,
$$c_1 G_1 + c_2 G_2 + \dotsc,$$
and I want to represent them in my Mathematica code. I represent graphs as adjacency matrices, e.g.</p>
<pre><code>{{0,1},{1,0}}
</code></pre>
<p>The next step is to write down linear combinations of these matrices. However, I want to imple... | Albert Retey | 169 | <p>I am not sure whether I understand correclty what you are after, but I think that using <code>UpValues</code> should help you solve your problem, e.g. this definition:</p>
<pre><code>Part[AdjMtx[data_], idcs___] ^:= Part[data, idcs]
</code></pre>
<p>will make it possible to access the matrix elements just as desir... |
1,776,260 | <p>After understanding the Cardano's formula for solving the depressed cubic (of the form $x^3+mx=n$, of course), I tried to find the solution of the equation $$x^3+6x=20.$$
After plugging into the formula
$$x=(n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}+(-n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}$$
where... | Siong Thye Goh | 306,553 | <p>\begin{align*}
&\left(10+\sqrt{108}\right)^{1/3}-\left(-10+\sqrt{108}\right)^{1/3}\\
&=\left(10+6\sqrt{3}\right)^{1/3}-\left(-10+6\sqrt{3}\right)^{1/3}\\
&=\left((1+\sqrt{3})^3\right)^{1/3}-\left((\sqrt{3}-1)^3\right)^{1/3}\\
&=(1+\sqrt{3})-(\sqrt{3}-1)\\
&=2
\end{align*}</p>
<p>The formula:
$$(... |
4,050,831 | <p>Suppose 40% of all seniors have a computer at home and a sample of 64 is taken. What is the probability that more than 30 of those in the sample have a computer at home?"</p>
<p>My attempt:</p>
<p>n=64</p>
<p>0.4x64=25.6</p>
<p>p=?</p>
<p>x=??</p>
<p>A>30=??</p>
<p>Don't have an idea of what equation would b... | BruceET | 221,800 | <p>The number <span class="math-container">$X$</span> of seniors in a random sample of size <span class="math-container">$n = 64,$</span> who will have computers at home has <span class="math-container">$X \sim \mathsf{Binom}(n = 64, p = 0.4).$</span> You seek
<span class="math-container">$P(X > 30) = 1 - P(X\le 30)... |
929,502 | <p>Here are two succinct statements of the 'same' question:</p>
<p><strong>Statement 1:</strong>
Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product function' $f:S\rightarrow \mathbb{R}^N; f(x_1,\dots,x_i)=\prod_ix_i$. There are many proofs that t... | Тимофей Ломоносов | 54,117 | <p>$\ln x$ is an increasing function. therefore $\arg \max\prod\limits_{i=1}^{x_n}x_i = \arg \max \ln \prod\limits_{i=1}^N x = \arg\max\sum\limits_{i=1}^n\ln x_i$. Once you've noticed it, it should be easier.</p>
|
2,249,020 | <p><a href="https://i.stack.imgur.com/L7PXf.jpg" rel="nofollow noreferrer">The Math Problem</a></p>
<p>I have issues with finding the Local Max and Min, and Abs Max and Min, after I find the Critical Point. How do I do this problem in its entirety? </p>
| Zain Patel | 161,779 | <p>Draw two position vectors, $\mathbf{v}_1$ and $\mathbf{v}_2$ with unit magnitude and at angles $\alpha, \beta$ to the positive $x$-axis. Then the angle between the two is $\alpha - \beta$ (assuming $\alpha > \beta$ w.l.o.g). But $\mathbf{v}_1 \cdot \mathbf{v}_2$ is the cosine of the angle between them. So $\cos (... |
3,988,808 | <p>I recently got into set theory and i was wondering what is the cardinality of a set of all finite sequences of natural numbers?
I know that it is N for natural numbers and 2^N is for real numbers but how can i prove it?</p>
| Ethan Bolker | 72,858 | <blockquote>
<p>Theorem: The set of all finite-length sequences of natural numbers is
countable.</p>
</blockquote>
<p>More is true: that follows from</p>
<blockquote>
<p>Theorem: (Assuming the axiom of countable choice) The union of
countably many countable sets is countable.</p>
</blockquote>
<p>You can find proofs ... |
3,988,808 | <p>I recently got into set theory and i was wondering what is the cardinality of a set of all finite sequences of natural numbers?
I know that it is N for natural numbers and 2^N is for real numbers but how can i prove it?</p>
| R.V.N. | 730,220 | <p>You might notice that your set is <span class="math-container">$\bigcup_{n\in\mathbb{N}}\mathbb{N}^n$</span>. I will assume you know that for every <span class="math-container">$n\in\mathbb{N}$</span>, <span class="math-container">$|\mathbb{N}^n|=|\mathbb{N}|$</span>. Then, for each <span class="math-container">$n\i... |
148,127 | <p>I am trying to get an analytic expression for this integral:</p>
<pre><code> Integrate[Sign[Cos[q]]/(q + 1), {q, 0, x}, Assumptions -> x > 0]
</code></pre>
<p>Mathematica gives the answer:</p>
<pre><code> Abs[Cos[x]] Log[1 + x] Sec[x]
</code></pre>
<p>However, when I compare it to the numerical integration... | zhk | 8,538 | <p>The functionality you are looking for is <code>Epilog</code>,</p>
<pre><code>c1 = ContourPlot3D[
x z + y z - x y z == 0, {x, 0, 2}, {y, 0, 2}, {z, 0, 2},
Epilog -> {Text[Style["c=0", 22], Scaled[{0.1, 0.0}]]}];
c2 = ContourPlot3D[
x z + y z - x y z == 0.2, {x, 0, 2}, {y, 0, 2}, {z, 0, 2},
Epilog -... |
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Igor Rivin | 11,142 | <p>Well, the Wikipedia gives <a href="https://en.wikipedia.org/wiki/Sylow_theorems#Sylow_theorems_for_infinite_groups" rel="nofollow noreferrer">an example of a Sylow theorem</a>, and there is more on this in notes <a href="https://people.brandeis.edu/%7Eigusa/Math131b/Sylow.pdf" rel="nofollow noreferrer">Sylow theorem... |
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Sh.M1972 | 44,949 | <p>The best reference for this subject is the book of Martyn Dixon: Locally finite groups and Sylow theory.</p>
|
1,613,863 | <p>How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$? </p>
| Jimmy R. | 128,037 | <p>If $n\ge 2$ (to be sure that $n+2\ge 4$ and $n\ge 2$ in the given binomial coefficients, otherwise see the comment below), you can write the LHS as $$\dbinom{n+2}{4}=\frac{(n+2)!}{4!(n+2-4)!}=\frac{(n+2)(n+1)n(n-1)(n-2)!}{4!(n-2)!}=\frac{(n+2)(n+1)n(n-1)}{24}$$ and the RHS as $$6\dbinom{n}{2}=6\frac{n!}{2(n-2)!}=6\... |
1,177,721 | <p>A fair $6$-sided die is rolled $6$ times independently. For any outcome, this is the set of numbers that showed up at least once in the different rolls. For example, the outcome is $(2,3,3,3,5,5)$, the element set is $\{2,3,5\}$. What is the probability the element set has exactly $2$ elements? how about $3$ element... | N. F. Taussig | 173,070 | <p>Exactly two numbers appear:</p>
<p>There are $\binom{6}{2}$ ways for two of the six numbers to appear. On each of the six rolls of the die, there are two possible outcomes, giving $\binom{6}{2} \cdot 2^6$ possible outcomes in which two of the six numbers appear. However, of these $2^6$ sequences, there are two in... |
11,994 | <p>Now that we get to see the SE-network wide list of "hot" questions, I am just shaking my head in disbelief. At the time I am writing this, the two hot questions from Math.SE are titled (get a barf-bag, quick)</p>
<ul>
<li><a href="https://math.stackexchange.com/q/599520/8348">https://math.stackexchange.com/q/599520... | user642796 | 8,348 | <p>These lists are automatically generated by a formula as described <a href="https://meta.stackexchange.com/a/61343/214632">here</a>.</p>
<p>According to <a href="https://meta.stackexchange.com/a/11604/214632">this</a> accepted MSO answer, in the past there was no intent to change the formula to account for the reput... |
3,260,911 | <p>I am currently struggling with the following exercise:</p>
<blockquote>
<p>Let <span class="math-container">$B$</span> be a Banach space and <span class="math-container">$C, D \subset B$</span> closed subspaces of <span class="math-container">$B$</span>.<br>
There is a <span class="math-container">$M \in ]0, \i... | D. A. | 671,115 | <p>I have proven it as follows:</p>
<p><span class="math-container">$B$</span> is a Banach space and <span class="math-container">$D, C \subseteq B$</span> are closed.
Thus <span class="math-container">$D$</span> and <span class="math-container">$C$</span> are themselves Banach spaces.
<span class="math-container">$C ... |
1,482,104 | <p>Let $X$ have a uniform distribution with p.d.f. $f(x) = 1$, $x$ is in $(0, 1)$, zero elsewhere.
Find the p.d.f. of $Y = -2 \ln X$.</p>
<p>I don't think this is a very difficult question, I just don't really understand what it is asking or where to start. Any help would be very much appreciated. Thank you! </p>
<p>... | UnknownW | 78,627 | <p>Let $f(x)=1_{(0,1)}(x)=F_{X}'(x)$. Note that
$$F_{Y}(y)=P(X\geq e^{-y/2})=1-P(X\leq e^{-y/2})=1-F_{X}(e^{-y/2}).$$
Then the density is
\begin{align}
F_{Y}'(y)&=(1-F_{X}(e^{-y/2}))'\\
&=-F_{X}'(e^{-y/2})\frac{\mathrm{d} }{\mathrm{d} y}(e^{-y/2})\\
&=-1_{(0,1)}(e^{-y/2})\cdot \left ( -\frac{1}{2}e^{-y/2} \... |
1,482,104 | <p>Let $X$ have a uniform distribution with p.d.f. $f(x) = 1$, $x$ is in $(0, 1)$, zero elsewhere.
Find the p.d.f. of $Y = -2 \ln X$.</p>
<p>I don't think this is a very difficult question, I just don't really understand what it is asking or where to start. Any help would be very much appreciated. Thank you! </p>
<p>... | mathreadler | 213,607 | <p>It was such a long time ago I did this, but this seems to work too. At least in my computer simulations. <strong>Maybe it is just a coincidence</strong> it works for this particular function, if that is the case, please show me why.</p>
<p>$Y = -2 \ln(X)$ means that if a sample of X is x, then the corresponding sam... |
1,482,104 | <p>Let $X$ have a uniform distribution with p.d.f. $f(x) = 1$, $x$ is in $(0, 1)$, zero elsewhere.
Find the p.d.f. of $Y = -2 \ln X$.</p>
<p>I don't think this is a very difficult question, I just don't really understand what it is asking or where to start. Any help would be very much appreciated. Thank you! </p>
<p>... | Upstart | 312,594 | <p>$ P(-2logX\le x)=P(logX\ge -x/2)=P(X\ge e^{\frac{-x}{2}})=\int_{e^{\frac{-x}{2}}}^1dt=1-e^{\frac{-x}{2}}$</p>
<p>$F_X(x)=1-e^{\frac{-x}{2}}$</p>
<p>hence $f_X(x)=\frac{1}{2}e^{\frac{-x}{2}}$<br>
which is the pdf of exponential with parameter $\frac{1}{2}$</p>
|
1,492,660 | <p>I'm teaching a course on discrete math and came across <a href="http://ac.els-cdn.com/0097316573900204/1-s2.0-0097316573900204-main.pdf?_tid=700c69c2-78e3-11e5-9825-00000aacb35e&acdnat=1445535573_08d35be15f0f7d7d939fc2800d9be60b" rel="nofollow">a paper related to the Hadwiger-Nelson problem</a>. The question ask... | Asaf Karagila | 622 | <p>There is no actual use of the axiom of choice. Since $\Bbb Q^2$ is a countable set, we can enumerate it and identify each equivalence class with the least-indexed point in that class.</p>
<p>Generally if $X$ can be well-ordered, then there is no need to use the axiom of choice in order to choose from equivalence cl... |
1,992,143 | <p>I'm trying to determine if $\sum \limits_{n=1}^{\infty} \sin(n\pi + \frac{1}{2n})$ absolutely converges or not.</p>
<p>Help me check it. I don't know how to do it. Advance thanks. :)</p>
| Jacky Chong | 369,395 | <p>Observe
\begin{align}
\sin\left(n\pi +\frac{1}{2n}\right) = (-1)^n\sin\frac{1}{2n}
\end{align}
then apply alternating series test. </p>
<p>It should be noted for large enough $n$ the terms $\sin\frac{1}{2n}$ decreases monotonically to zero. </p>
<p>Hence the series converges conditionally.</p>
<p>Edit: Let us con... |
749,714 | <p>Does anyone know how to show this preferable <strong>without</strong> using modular</p>
<p>For any prime $p>3$ show that 3 divides $2p^2+1$ </p>
| Andreas Caranti | 58,401 | <p>If you want to avoid modular arithmetic, you can easily do the explicit calculations, distinguishing the two cases</p>
<ul>
<li>$p = 1 + 3 k$, then $$2 p^{2} + 1 = 2 (1 + 6k + 9 k^{2}) + 1 = 3 (1 + 4k + 6 k^{2})$$</li>
<li>$p = 2 + 3 k$, then $$2 p^{2} + 1 = 2 (4 + 12k + 9 k^{2}) + 1 = 3 (3 + 8k + 6 k^{2})$$</li>
<... |
4,498,498 | <p>Lets say you have two sequences of non negative integers each of length <span class="math-container">$n$</span>.</p>
<p>ie <span class="math-container">$(a_1,a_2,...,a_n)$</span> and <span class="math-container">$(b_1,b_2,...,b_n)$</span> such that
<span class="math-container">$\max(a_i) < k$</span>
and
<span cla... | AlvinL | 229,673 | <p>Not an answer but an elaboration on P. Quinton's remark.</p>
<p>A pair of sequences <span class="math-container">$a$</span> and <span class="math-container">$b$</span> is solvable if and only if the pair of sequences
<span class="math-container">$$(\min (a_1,b_1),\ldots, \min(a_n,b_n))\quad\mbox{and}\quad (\max (a_1... |
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | copper.hat | 27,978 | <p>By inspection we see that $A (1,1,1,1)^T = 2 (1,1,1,1)^T $.</p>
<p>Similarly, $A (1,0,-1,0)^T = 0 $, $A (0,1,0,-1)^T = 0 $ and
$A(1,-1,1,-1)^T = 2 ((1,-1,1,-1)^T$.</p>
<p>All four vectors are linearly independent, hence constitute a
basis of eigenvectors.</p>
|
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | Community | -1 | <p>Developing $|A-\lambda I|$, on the first row you get</p>
<p>$$
(1-\lambda)\begin{vmatrix}
1-\lambda & 0 & 1\\
0 & 1-\lambda & 0\\
1 & 0 & 1-\lambda\\
\end{vmatrix}+
\begin{vmatrix}
0 & 1-\lambda & 1\\
1 & 0 & 0\\... |
248,706 | <p>Let $X$ be a compact connected manifold. Since $\mathbb T^1$ is an Eilenberg-MacLane space $K(\mathbb Z,1)$, it follows that for every morphism $\varphi\colon\pi_1(X)\to\pi_1(\mathbb T^1)$ there is a continuous map $f\colon X\to\mathbb T^1$ such that $\varphi$ coincides with the induced morphism $f_*$.</p>
<p>Now a... | Qiaochu Yuan | 290 | <p>There is no need to assume that $X$ is compact, connected, or a manifold in your first claim. Anyway, here is some context in which to put this question. A morphism $G \to S^1$ of Lie groups gives a map $BG \to BS^1 \cong B^2 \mathbb{Z}$ of classifying spaces, and hence a cohomology class in $H^2(BG, \mathbb{Z})$. I... |
15,205 | <p>I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single demonstration, studying the basics every time if I don't remember them. And this makes it impossible for me to prepare ... | guest | 11,881 | <p>I am going to assume that you are teaching a calculus "helper" versus the entire physics class. Your initial statements don't match that. But then all your content described is math, not physics. And also 50 minutes per week sounds rather light for a whole class. [If the converse is the case, I would spend your ... |
2,231,003 | <p>I am trying to prove that the function $f_a(x) = e^{-\frac{1}{(x-a)}}$ is differentiable for all $x>a$. However, I do not know how to show $|\frac{e^{-\frac{1}{(x-a)}}-e^{-\frac{1}{(p-a)}}}{x-p} - (-\frac{e^{-\frac{1}{(p-a)}}}{(p-a)^2})| < \epsilon$ if $0<|x-p|<\delta$. </p>
<p>My next idea was to apply... | Yes | 155,328 | <p>You were almost there! Note that $f_{a}: x \overset{g}{\mapsto} \frac{-1}{x-a} =: y \overset{h}{\mapsto} e^{y}$. You also know that both $g$ and $h$ are differentiable. So by the chain rule $f_{a} = h \circ g$ is also differentiable (its proof is contained as a special case in the proof of chain rule). </p>
<p>The ... |
628,682 | <p>As both a programmer and a math student, I am trying to come up with a fool-proof way to handle errors from subtractive cancellation caused by trying to evaluate $x-y$, where x,y are extended (long double) precision floating-point numbers. (Obviously, if x is very close to y, this causes problem.) I found two equiva... | Mike Warren | 78,453 | <p>I think my error lies in getting from $x<1-\sqrt{y}$ to $x-y<1-2\sqrt{y}$. Since I have already imposed the condition that x,y>0, it would then follow that $x,y\in [0,1]$. I then went from $x-y<1-2\sqrt{y}$ to $y>1$, <strong>which contradicts the previous statement</strong>.</p>
|
3,141,618 | <p>The exercise is:</p>
<blockquote>
<p>Show that if <span class="math-container">$A \subset \mathbb{R} $</span> is bounded and <span class="math-container">$ A \neq \varnothing $</span> then <span class="math-container">$sup(A)=max(\overline{A} ).$</span></p>
</blockquote>
<p>Now, I wanted to ask you <strong>wheth... | Henno Brandsma | 4,280 | <p>First show that <span class="math-container">$\sup(A)$</span> (which exists by boundedness of <span class="math-container">$A$</span> and completeness of <span class="math-container">$\mathbb{R}$</span>) is in <span class="math-container">$\overline{A}$</span>. This follows from the definition of sup and the fact th... |
977,446 | <p>Prove that $A\cap B = \emptyset$ iff $A\subset B^C$. I figured I could start by letting $x$ be an element of the universe and that $x$ is an element of $A$ and not an element of $B$. </p>
| AlexR | 86,940 | <p>Recall some definitions: Let $\Omega$ denote the universe (At least $A\cup B$).
$$B^C = \{ x \in\Omega | x\notin B\} \\
M\subset C :\Leftrightarrow \forall m\in M: m\in C \\
A\cap B := \{x\in\Omega | x\in A \wedge x\in B\}$$
Now plug this in: $A\subset B^C \Rightarrow A\cap B = \emptyset$ starts with any $x\in \Omeg... |
3,489,347 | <p><strong>Is there a simple way to characterize the functions in <span class="math-container">$C^\infty((0,1])\cap L^2((0,1])$</span>?</strong></p>
<p>That is, given a function <span class="math-container">$f(t)\in C^\infty((0,1])$</span>, is there a necessary/sufficient condition I can check to see if it's square in... | ZAF | 609,023 | <p><span class="math-container">$N \in \mathbb{N}$</span> ?</p>
<p><span class="math-container">$10N\log(N)> 2N^2$</span> </p>
<p>If and only if</p>
<p><span class="math-container">$5 \log(N)> N$</span> </p>
<p>If and only if </p>
<p><span class="math-container">$e^{5\log(N)} > e^{N}$</span></p>
<p>If an... |
3,893,440 | <p>Suppose we have <span class="math-container">$4$</span> books on Math, <span class="math-container">$5$</span> books on English and <span class="math-container">$6$</span> books on History. In how many ways you can put them on your bookshelf if you want :- <br/>
<span class="math-container">$1)$</span> The first boo... | Scott Hahn | 714,928 | <p>These are <span class="math-container">$4$</span> separate homework problems, right? In total we have <span class="math-container">$15$</span> books.</p>
<p><span class="math-container">$1)$</span> <span class="math-container">$4 * 14!$</span> (First choose any math book, then for each choice, we can order the remai... |
3,893,440 | <p>Suppose we have <span class="math-container">$4$</span> books on Math, <span class="math-container">$5$</span> books on English and <span class="math-container">$6$</span> books on History. In how many ways you can put them on your bookshelf if you want :- <br/>
<span class="math-container">$1)$</span> The first boo... | Anonymous | 772,237 | <p>Here is the solution for the <span class="math-container">$3$</span>rd part .</p>
<p>Assume the <span class="math-container">$4$</span> Math and <span class="math-container">$5$</span> English books to be in a line, as in each arrangement they have to be consecutively placed. Take this to be a block of <span class="... |
1,067,051 | <p>How can I find the point of intersection of <span class="math-container">$y=e^{-x}$</span> and <span class="math-container">$y=x$</span> ?</p>
<p><a href="https://i.stack.imgur.com/VoX32.png" rel="nofollow noreferrer">Here's the graph</a></p>
| Joel | 201,006 | <p>First:</p>
<p>There is no closed form for the solution. You can get an arbitrarily good approximation with root finding.</p>
<p>Have you learned about cobweb diagrams? That's one of the simplest versions of root finding. Otherwise, perhaps Newton's method? I don't want to give too much away (looks like a homew... |
1,365,489 | <p>What is the value of the following expression?</p>
<p>$$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$</p>
| No_way | 244,997 | <p>It is $2\sqrt{5}$. Note that $(17\sqrt{5}+38)^{\frac{1}{3}}=2+\sqrt{5}$. </p>
|
126,739 | <p><strong>I changed the title and added revisions and left the original untouched</strong> </p>
<p>For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the factorials in the denominator of the binomial theorem;
$$\sum _{k=1}^{\infty } \frac{1}{k!} \... | Douglas Zare | 2,954 | <p>In the comments, quid showed that $\sum_{k=0}^\infty \frac{1}{(k+m)k!} = (-1)^m\bigg((m-1)! - !(m-1)e \bigg)$ where $!a = a! \sum_{k=0}^a \frac{(-1)^k}{k!}$ is the number of <a href="http://oeis.org/wiki/Number_of_derangements" rel="nofollow">derangements</a> in the symmetric group on $a$ objects. For example, $\sum... |
1,288,584 | <p>If <span class="math-container">$f(x)=f(x_0)+f'(x_0)(x-x_0)+\ldots+\frac{f^{(n-1)}(x_0)}{(n-1)!}(x-x_0)^{n-1}+\frac{f^{(n)}(\xi(x))}{n!}(x-x_0)^n,$</span> prove that <span class="math-container">$x \rightarrow f^{(n)}(\xi(x)) $</span> is continuous on <span class="math-container">$[x_0-\beta, x_0+\beta]$</span>, if... | DeepSea | 101,504 | <p>Let $g(x) = f^{(n)}\left(\xi(x)\right)$, and let $t \in[x_0-\beta,x_0+\beta]$, we show $g$ is continuous at $x = t$, i.e. $\displaystyle \lim_{x \to t} g(x) =g(t)$. We have: $x_0 < \xi(t) < t$, and $x_0 < \xi(x) < x$. We prove: $\displaystyle \lim_{x\to t} \xi(x) = \xi(t)$. To this end, assume WLOG:
$x_0... |
1,975,018 | <p>I have an equation: </p>
<p>$$y = \Bigl(x + f(x^2 - 1)\Bigr)^{1/2},$$
(so a square root function.) I am asked to find $dy/dx$ when $x = 3$, given that $f(8) = 0$, and $f'(8) = 3$.</p>
<p>I apply the chain rule as I usually do for this problem, derivative of the inside times derivative of the outside. I get:</p>
... | Claude Leibovici | 82,404 | <p>Consider $$y = \Bigl(x + f[t(x)]\Bigr)^{1/2}$$ An easy way could be logarithmic differentiation $$y = \Bigl(x + f[t(x)]\Bigr)^{1/2}\implies \log(y)=\frac 12 \log\Bigl(x + f[t(x)]\Bigr)$$ $$\frac {y'}y=\frac 12 \frac{1+\frac{df}{dt}\times \frac{dt}{dx}}{x + f[t(x)]}\implies y'= \frac{1+\frac{df}{dt}\times \frac{dt}{d... |
1,975,018 | <p>I have an equation: </p>
<p>$$y = \Bigl(x + f(x^2 - 1)\Bigr)^{1/2},$$
(so a square root function.) I am asked to find $dy/dx$ when $x = 3$, given that $f(8) = 0$, and $f'(8) = 3$.</p>
<p>I apply the chain rule as I usually do for this problem, derivative of the inside times derivative of the outside. I get:</p>
... | operatorerror | 210,391 | <p>Hint: Rewrite $f(x^2-1)$ as $(f\circ g )(x)$ where $g(x)=x^2-1$, and then take the derivative of the composition using the chain rule. </p>
|
1,866,931 | <p>I would like to see a proof to this fact.</p>
<blockquote>
<p>If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there holds
$$
\|B-A\| \|A^{-1}\| <1,
$$
we have that B is invertible.</p>
</blockquote>
<p>Moreov... | Anon | 245,264 | <p>Your formulation is incorrect. You're doing the right steps, but you're writing them down wrong, the logical meaning of what you write is different from what you mean. Here's proper formulation:</p>
<p>Let $R$ be a relation on $A$ such that $R \circ R \subseteq R$.</p>
<p>Let $x, y, z \in A$ such that $(x, y), (y,... |
755,571 | <p>$$a_n=3a_{n-1}+1; a_0=1$$</p>
<p>The book has the answer as: $$\frac{3^{n+1}-1}{2}$$</p>
<p>However, I have the answer as: $$\frac{3^{n}-1}{2}$$</p>
<p>Based on:</p>
<p><img src="https://i.stack.imgur.com/4vJrQ.png" alt="enter image description here"></p>
<p>Which one is correct?</p>
<p>Using backwards substit... | AsdrubalBeltran | 62,547 | <p>Ok your answer is correct, but you have the formula for the $(n-1)$-term, if you need the formula for the $n$-term, then note that, according to recurrence</p>
<p>$$a_n=3a_{n-1}+1$$</p>
<p>You have that $$a_{n-1}=\frac{3^n-1}{2}$$ then:
$$a_n=3\cdot\frac{3^n-1}{2}+1=\frac{3^{n+1}-1}{2}$$</p>
<p>God bless</p>
|
1,217,175 | <p><strong>Here's the question:</strong></p>
<p>Is the following true or false?</p>
<p>There is a function $f: \mathbb R \to \mathbb R$ that satisfies the following condition:</p>
<p>For every $a \in \mathbb R $ and $ \epsilon \gt 0 $ there is $\delta \gt 0$ such that $\left| f(x)-f(a) \right| \lt \epsilon \implies ... | Ali Caglayan | 87,191 | <p>Let $x=a+bi$ then $\sqrt{x}=\sqrt{a+bi}=-1$. Then square both sides to get $a+bi=1$. Thus $x$ must be a 'real' number and not complex. Now if $a=1$, taking the <em>positive</em> square root, which is what $\sqrt{.}$ means, will give you that $\sqrt{1}=-1$. Which is not strictly true when using the radical sign.</p>
... |
3,848,929 | <p>The question is entirely in the title. I need this result as part of a much bigger question. If this is possible, please give me an example of such <span class="math-container">$u$</span> and <span class="math-container">$v$</span>.</p>
<p><strong>My progress:</strong>
Let <span class="math-container">$u=2k+1$</span... | metamorphy | 543,769 | <p>Let <span class="math-container">$u,v$</span> be coprime and <span class="math-container">$u+v=(u-v)d$</span>. Then <span class="math-container">$(d-1)u=(d+1)v$</span>.</p>
<p>Thus, <span class="math-container">$u\mid(d+1)v$</span>, hence <span class="math-container">$u\mid(d+1)$</span> and <span class="math-contain... |
3,848,929 | <p>The question is entirely in the title. I need this result as part of a much bigger question. If this is possible, please give me an example of such <span class="math-container">$u$</span> and <span class="math-container">$v$</span>.</p>
<p><strong>My progress:</strong>
Let <span class="math-container">$u=2k+1$</span... | Haran | 438,557 | <p>Assume <span class="math-container">$u-v>2$</span>. Let <span class="math-container">$p$</span> be an odd prime divisor of <span class="math-container">$u-v$</span>. Then, <span class="math-container">$p \mid (u+v)$</span> as well. Adding and subtracting, we can see that <span class="math-container">$p$</span> di... |
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