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 |
|---|---|---|---|---|
2,648,526 | <p>I really cannot figure this question out. Can anyone help me please!?</p>
<blockquote>
<p>Prove that the length of the median $m_a$ of obtuse triangle $△ABC$ with the obtuse $∠CAB$ is smaller than $\dfrac{1}{2}|BC|$.</p>
</blockquote>
<p>Thank you very much!</p>
| Michael Rozenberg | 190,319 | <p>Let $\Phi$ be the circle with diameter $BC$.</p>
<p>Since $\measuredangle BAC>90^{\circ},$ we obtain that $A$ is placed inside the circle.</p>
<p>Now, let $M$ be the midpoint of $BC$ with the ray $MA\cap \Phi=\{A_1\}.$</p>
<p>Thus, $$\frac{1}{2}BC=A_1M>AM=m_a$$ and we are done!</p>
|
3,051,281 | <p>The question is:</p>
<p>Let <span class="math-container">$n \ge 2$</span> be an integer and consider a uniformly random permutation
(<span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, . . . , <span class="math-container">$a_n$</span>) of the set (1, 2, . . . , n). </p>
<p>For ea... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<ul>
<li><p>Choose particular value of <span class="math-container">$x$</span>, for example choose <span class="math-container">$x=0$</span>. You should be able to solve one of the unknown directly.</p></li>
<li><p>Can you choose another value of <span class="math-container">$x$</span> to make <span clas... |
3,051,281 | <p>The question is:</p>
<p>Let <span class="math-container">$n \ge 2$</span> be an integer and consider a uniformly random permutation
(<span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, . . . , <span class="math-container">$a_n$</span>) of the set (1, 2, . . . , n). </p>
<p>For ea... | Shubham Johri | 551,962 | <p>If any one of <span class="math-container">$\lambda_1,\lambda_2$</span> is non-zero, the <span class="math-container">$LHS$</span> will be periodic with period <span class="math-container">$2\pi$</span> while the <span class="math-container">$RHS$</span> is not. Therefore, <span class="math-container">$\lambda_{1,2}... |
1,164,471 | <p>I'm having trouble with the following problem:</p>
<p>A man found that $3$ out of $10$ inspected bottles were defective. What is the probability that the $2$ first defective bottles were found in the first $7$ inspected bottles? The probability that a bottle is defective is $0.1988$.</p>
<p>Let $A$ be the event th... | Krish | 177,430 | <p>Define a map $\phi:U(n) \to Gal(Q(\xi_n)/Q), a \mapsto \sigma_a$ where $\sigma_a:Gal(Q(\xi_n)/Q) \to Gal(Q(\xi_n)/Q)$ is defined by $\xi_n \to \xi_n^a.$ (Here $U(n) = (\mathbb Z/n\mathbb Z)^{\times}$ and $\xi_n$ is a primitive root of unity.) Then $\phi$ will be an isomorphism.</p>
<p><em>Note:</em> See <a href="h... |
453,502 | <p>Let $I$ be a generalized rectangle in $\Bbb R^n$ </p>
<p>Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$</p>
<p>Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$. </p>
<hr>
<p>My idea is that</p>
<p>For $(\impliedby)$</p... | Pedro | 23,350 | <p>One direction is obvious. If $f$ is identically $0$, the integral is zero. Now suppose $f$ is continuous but <strong>not</strong> identically zero. Then there exists a point in $I$ such that $f(\xi)>0$. By continuity there exists a neighborhood of $\xi$ where $f(x)>0$ whenever $x\in(\xi-\delta,\xi+\delta)$. Ca... |
2,166,847 | <p>Let $$v_n=\dfrac 1 {n+1}\sum_{k=0}^n \dfrac 1 {k+1}$$
We wanna study the sum $$S=\sum_{n=0}^{+\infty}(-1)^n v_n$$</p>
<p>The problem says we should first find $\omega(x)$ s.t.
$$v_n=\int_0^1 x^n\omega(x)dx$$
Then we'll have $S=\int_0^1\dfrac {\omega(x)} {1+x}dx$, but I can't find such $\omega(x)$. What's the idea ... | Jack D'Aurizio | 44,121 | <p>After we get
$$ S = -\int_{0}^{1}\frac{\log(1-x)}{1+x}\,dx = -\int_{0}^{1}\frac{\log x}{2-x}\,dx =-\left.\frac{d}{d\alpha}\int_{0}^{1}\frac{x^{\alpha}\,dx}{2-x}\,\right|_{\alpha=0^+}$$
we also have
$$ S = \text{Li}_2\left(\frac{1}{2}\right) = \color{red}{\frac{\pi^2}{12}-\frac{\log(2)^2}{2}}$$
by the <a href="https:... |
1,934,268 | <p>Let's say I have an arbitrary matrix $A$. </p>
<p>An eigenvector of $A$ would be a vector $\vec{x}$, such that </p>
<ol>
<li>$A\vec{x} = \lambda \vec{x}$</li>
<li>$\vec{x} \in N(A-\lambda I)$ i.e. $(A - \lambda I)\vec{x} = \vec{0}$</li>
</ol>
<p>where $\lambda$ is an eigenvalue of $\vec{x}$</p>
<hr>
<p>Now are ... | egreg | 62,967 | <p>The answer to your title question is "just some", because there is a single exception: the null vector.</p>
<p>By definition, an eigenvalue of $A$ relative to $\lambda$ is a <em>nonzero</em> vector $x$ such that $Ax=\lambda x$ (forgive me if I don't use arrows for vectors). Now
$$
Ax=\lambda x
\iff
Ax-(\lambda I)x=... |
3,435,511 | <p>I'm looking to write down formally that a multiset of elements contains at least two elements that differs in value.
e.g.,
S1 = {1,1,1,1,1,1} and S2={1,1,1,1,0,1}
S1 has all identical elements, S2 has at least two elements that differs in value.</p>
| Robert Israel | 8,508 | <p>Write <span class="math-container">$$\ln(1-e^{-k}) = - \sum_{j=1}^\infty e^{-jk}/j$$</span>
and take <span class="math-container">$k = x + t$</span> so your integral becomes
<span class="math-container">$$ \sum_{j=1}^\infty e^{-jx} \int_0^\infty \frac{\sqrt{t(t+2x)}}{j(x+t)} e^{-jt} \; dt$$</span>
Now <span class="m... |
4,030,877 | <p><strong>Question:</strong> Ten different candies are given to three children <code>A</code>, <code>B</code> and <code>C</code>. each child has at least one. How many different ways are there?</p>
<p>I use two different methods to solve this problem:</p>
<pre><code> f[1, n_] := n;
f[n_, 1] := 1;
f[n_, m_] ... | Ben | 887,799 | <p><span class="math-container">$$\sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x}$$</span></p>
<p>This means that the value of the right is exactly equal to that on the right and most importantly the right converges if and only if the left converges.</p>
<p>For any x, <span class="math-container">$\frac{1-x^{n+1}}{1-x} = \fra... |
3,091,464 | <p>If I had an array of 7 numbers and I wanted all numbers to be equally spaced within it but needed to start at 5 and end at 35 how would I do this?</p>
<p>For instance if I were given:</p>
<ul>
<li>Numbers in array: <strong>7</strong></li>
<li>Largest number in array: <strong>35</strong></li>
<li>Smallest number in... | RRL | 148,510 | <p>Correct.</p>
<p>As the extension is bounded, where <span class="math-container">$|\hat{f}(x)| \leqslant M$</span> on <span class="math-container">$[0,1]$</span>, we have as <span class="math-container">$\epsilon \to 0$</span>,</p>
<p><span class="math-container">$$\left|\int_0^1\hat{f} - \int_\epsilon^{1-\epsilon}... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Robert Mastragostino | 28,869 | <p>You probably won't find one book that covers everything, but in all honesty if you're a good math student you could probably skip through trigonometry with ease. I'd recommend <a href="http://books.google.ca/books?id=Gs2pHYioDbQC&printsec=frontcover&dq=algebra+demystified&source=bl&ots=cxxr2HGerP&... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Martin Peters | 185,067 | <p>Have a go at the book <em>Mathematical omnibus. Thirty lectures on classic mathematics</em> by Dmitry Fuchs and Sergei Tabachnikov. This book will engage your mind and point you to very interesting mathematics.</p>
<p>Another good book to try out is this one: <a href="http://imaginary.org/sites/default/files/5to15_... |
61,697 | <p>The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:</p>
<p>Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x_0$ such that $\phi_{\ast}h(x)=h(x_0)+\omega_0(\phi_{\as... | agt | 12,617 | <p>Warning: I have posted this as an answer and not as comment, not to gain in reputation, but just in order to have enough space to write to Michael what I had understood of his answer that has been very much useful to me.</p>
<p>Let $(M_0,\omega_1)$ and $(M_1,\omega_1)$ be symplectic manifolds of the same dimension.... |
2,970,773 | <p>Let's assume I am given a positive integer <span class="math-container">$n$</span>, as well as an upper limit <span class="math-container">$L$</span>.</p>
<p>How could one find all, or at least one, possible solutions for <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that ... | user87690 | 87,690 | <p>The space is assumed to be regular, so if <span class="math-container">$\{B_n\}_{n ∈ ℕ}$</span> is a basis, then for every <span class="math-container">$x ∈ B_m$</span>, there exists <span class="math-container">$n$</span> such that <span class="math-container">$x ∈ B_n ⊆ \overline{B_n} ⊆ B_m$</span>. Nobody claims ... |
3,412,217 | <p>The category of abelian groups <span class="math-container">$\mathbb{Ab}$</span> has a monoidal (closed) structure <span class="math-container">$(\otimes, \mathbb{Z})$</span>. Moreover, it is monadic over the category of sets via the free abelian group monad <span class="math-container">$$\mathbb{Z}[\_]: \text{Set} ... | Z Ahmed | 671,540 | <p>If GCD od <span class="math-container">$a$</span> and <span class="math-container">$b$</span> is 1.
Let the pairs of integers <span class="math-container">$(x_0,y_0)$</span>, be the first solution of <span class="math-container">$ax+by=n~(*)$</span>. Then
Let <span class="math-container">$x=x_0+bm, y=y_0-am$</span>... |
2,265,819 | <p>I need to parameterize the intersection of $$4x^2 + y^2 + z^2 = 9\tag{1}$$ and $$z=x^2+y^2\tag{2}$$. </p>
<p>First, I'll solve (2) for $y^2$ and substitute the result into (1):</p>
<p>$$3x^2+z+z^2 = 9 \tag{3}$$</p>
<p>Next, I'll make the substitution $u=\sqrt{3}x$, such that we can complete the square in (3) by a... | Yiorgos S. Smyrlis | 57,021 | <p><strong>Step 1.</strong> Clearly $\mathrm{Im}\,f(x+iy)\ne 0$, whenever $y\ne 0$. In particular,
$\mathrm{Im}\,f(z)$ maintains sign in the upper half plane and the lower half plane, and it has different signs in each half plane. Without loss of generality, $\mathrm{Im}\,f(x+iy)> 0$, if $y>0$, and $\mathrm{Im}\... |
4,418,518 | <p><span class="math-container">$X$</span> is Banach Space and <span class="math-container">$A\subseteq X$</span>.
<span class="math-container">$A$</span> is dense <span class="math-container">$G_{\delta}$</span> set. We have to show that <span class="math-container">$A-A=X$</span>.</p>
<p>Since <span class="math-conta... | Gerd | 960,044 | <p>As <span class="math-container">$A$</span> is a dense <span class="math-container">$G_\delta$</span> set the complement <span class="math-container">$X\setminus A$</span> is of first category. Let <span class="math-container">$x_0 \in X$</span> and set <span class="math-container">$h:X \to X$</span>, <span class="ma... |
4,448,525 | <p>To do a proof by induction, it is necessary to prove</p>
<p>(1) the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=1$</span></p>
<p>(2) if the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=k$</span>, then <span c... | Robert Shore | 640,080 | <p>Fundamentally, a proof by induction is using the well-ordering property of the natural numbers, i.e., the fact that every <em>non-empty</em> set of natural numbers has a least element. And it can always be thought of as a proof by contradiction. You’re assuming the set of counterexamples of <span class="math-conta... |
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| E.H.E | 187,799 | <p>$$\frac{\log 4}{\log b}=\frac{\log(4^{1/4})^4}{\log(b^{1/b})^b}=\frac{4}{b}\frac{\log\sqrt[4]{4}}{\log\sqrt[b]{b}}$$
so
$$\frac{\log 4}{\log b}\neq \frac{4}{b}\quad{\text{(except when b=4 and b=2})}$$</p>
|
3,543,060 | <p>The ODE is defined on <span class="math-container">$[0,b]$</span> with Neumann boundary conditons.
<span class="math-container">$$y''(x)=\frac{c_1y(x)}{c_2+y(x)}$$</span>
<span class="math-container">$$y'(0)=0; y'(b)=0$$</span></p>
<p>How to solve the above ODE? Any help is appreciated!</p>
| user577215664 | 475,762 | <p><span class="math-container">$$y''(x)=\frac{c_1y(x)}{c_2+y(x)}$$</span>
Multiply both sides by <span class="math-container">$2y'$</span>:
<span class="math-container">$$2y'y''(x)=\frac{2c_1y(x)y'}{c_2+y(x)}$$</span>
<span class="math-container">$$(y'^2)'=\frac{2c_1y(x)y'}{c_2+y(x)}$$</span>
Integrate:
<span class="m... |
29,601 | <p>I know the order of the group is the number of elements in the set. For example the group of $U_{10}$ (units of congruence class of 20) has order 4. </p>
<p>Major Edit, kinda changed the question.
Lets say my element $a$ has a finite order $n$. Then what is the order of $a, a^2, a^3...a^{11}$?</p>
| Fredrik Meyer | 4,284 | <p>The order of an element in a group is the order of the subgroup it generates. Equivalently, it is the least integer n such that $a^n$ is the identity. If the order of the group is n, then the order of any element in the group actually divides n (this is Lagranges theorem).</p>
|
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | vi pa | 58,013 | <pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[decimal_]:=If[StringCases["11"][IntegerString[decimal,2]]=={},False,True]
consecutiveOnes/@declist
</code></pre>
<p><a href="https://i.stack.imgur.com/Qp43v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qp43v.png" alt="enter i... |
1,032,874 | <p>Prove that $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cdots + n^3$ for every $n \in \mathbb{N}$.</p>
<p>Proof. We will use mathematical induction.
If $n = 1$, then we have $(1)^2= 1^3 = 1$. We must show that $S_n$ implies $S_{n+1}$. Assume that for $n \in \mathbb{N}$, the statement $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cd... | TonyK | 1,508 | <p>Both sides are polynomials of degree $4$. Both sides take the values $0,1,9,36,100$ for $n=0,1,2,3,4$. So they are the same polynomial!</p>
|
1,032,874 | <p>Prove that $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cdots + n^3$ for every $n \in \mathbb{N}$.</p>
<p>Proof. We will use mathematical induction.
If $n = 1$, then we have $(1)^2= 1^3 = 1$. We must show that $S_n$ implies $S_{n+1}$. Assume that for $n \in \mathbb{N}$, the statement $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cd... | Martín-Blas Pérez Pinilla | 98,199 | <p>Interesting graphical proof with applet in <a href="https://www.geogebratube.org/student/m37497" rel="nofollow">The Sum of the Cubes of the First N Natural Numbers</a>.</p>
|
2,668,538 | <p>Firstly I'm aware of the proofs/reasons regarding .999...=1. I'm not asking for anyone to reference or reiterate them but rather to look at my proof in isolation and help me understand my own mistakes and fallacies.</p>
<p>Another disclaimer I suppose; it's difficult to call this 'proof' my 'own' as it's extremely ... | Andrew Li | 344,419 | <p>The very assertion that:</p>
<p>$$\dfrac{1}{0.\bar 9} = 1 + 0.\bar 01$$</p>
<p>Proves that $1 = 0.\bar 9$. This is because:</p>
<p>$$0.\bar 01 = 0$$</p>
<p>Conceptually speaking, there is no way to stick a $1$ at the end of an infinite number of $0$s because there is no end to an infinite number of $0$s. There w... |
4,630,145 | <p>We need to determine all <span class="math-container">$\mathbb{Q}$</span>-homomorphisms from <span class="math-container">$\mathbb{Q}(\sqrt[5]2)$</span> to <span class="math-container">$\mathbb{Q}(\sqrt[5]2, \omega)$</span>, where <span class="math-container">$\omega=e^{2\pi i/5}$</span>.</p>
<p>I can figure it out ... | simo210 | 1,142,316 | <p>First, notice that <span class="math-container">$\mathbb{Q}(\sqrt[5]{2},\omega)$</span> is the splitting field over <span class="math-container">$\mathbb{Q}$</span> of the polynomial <span class="math-container">$f(X)=X^5-2$</span>.</p>
<p>In general, it's a well known fact that the set of <span class="math-containe... |
1,948,163 | <p>I don't know how to differentiate this function. $$u=\left(\frac{1}{t}-\frac{1}{\sqrt{t}}\right)^{2}$$</p>
<p>Should I use the quotient rule or just the power chain rule?</p>
| yngabl | 111,900 | <p>You want $\frac{du}{dt}$ so set $$p=\frac{1}{t}-\frac{1}{\sqrt{t}}$$ then $u=p^2$ and $\frac{du}{dt}=2p\frac{dp}{dt}$. Calculate $\frac{dp}{dt}$, put back with $p=\sqrt{u}$ and you will find the answer.</p>
|
725,746 | <p>I am not sure I understand the $N - \epsilon$ method for proving the equality of a limit.</p>
<p>I have a past mid-semester exam question that has:
$$\lim \limits_{x \to 1} (x^2 - 4x) = -3$$</p>
<p>Now it seems I want to take the $-3$ over $\rightarrow$ $|x^2 - 4x + 3| \lt \epsilon$ $$\text{ }\text{ }\text{ }\t... | DeepSea | 101,504 | <p>You want to estimate $|x - 3|$. One way to get an upper bound for this is to use the triangle inequality: $|x - 3| =|(x - 1) + 2| \longleftarrow |x - 1| + |2| = |x - 1| + 2$. So if you let $|x - 1| \lt 1$, then $|x - 3| \lt 1 + 2 = 3$. So $|x - 3| \cdot |x - 1| \lt 3|x - 1|$. And you need $3|x - 1| \lt e$. All you n... |
725,746 | <p>I am not sure I understand the $N - \epsilon$ method for proving the equality of a limit.</p>
<p>I have a past mid-semester exam question that has:
$$\lim \limits_{x \to 1} (x^2 - 4x) = -3$$</p>
<p>Now it seems I want to take the $-3$ over $\rightarrow$ $|x^2 - 4x + 3| \lt \epsilon$ $$\text{ }\text{ }\text{ }\t... | davidlowryduda | 9,754 | <p>Continue reasoning. The values of $x$ you are restricting to are somewhere near $1$. Let's say now and forever that we will demand the distance from $1$, which I'll call $\delta$, is at most $1$. (I'm not quite sure what you mean by $N$ - I suspect you would call my $\delta = \frac{1}{N}$). I do this so that I can s... |
3,525,017 | <p>a) Show that the generating function by length for binary strings where every block of 0s has length at least 2, each block of ones has length at least 3 is:</p>
<p><span class="math-container">$$\frac{(1-x+x^3)(1-x+x^2)}{1-2x+x^2-x^5}$$</span></p>
<p>b) Give a recurrence relation and enough initial conditions to ... | RobPratt | 683,666 | <p>Let <span class="math-container">$a_n$</span> be the number of strings that start with <span class="math-container">$0$</span>, and let <span class="math-container">$b_n$</span> be the number of strings that start with <span class="math-container">$1$</span>. Then <span class="math-container">$a_0=b_0=1$</span>, <s... |
95,341 | <p>I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).</p>
<p>Is it possible to extend the proof also to polynomials
$P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$ ?</p>
<p>Thanks!</p>
| Marc Palm | 10,400 | <p>Let $X_{j}$ be elements of $SL(n, \mathbb{C})$ with entries ${X_{j}^{k,l}}$.</p>
<p>Express the polynomial $P: SL(n,\mathbb{C})^N \rightarrow \mathbb{C}$ as a polynomial $Q: \mathbb{C}^{n \cdot N} \rightarrow \mathbb{C}$ in variables ${X_{j}^{k,l}}$</p>
<p>Similar express the polynomial
$$ Q''(X_1, \cdots, X_n) =... |
1,468,070 | <p>Let <span class="math-container">$\{a_n\}_1^\infty$</span> and <span class="math-container">$\{b_n\}_1^\infty$</span> be two sequences in <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$\forall n \in \mathbb{N}$</span>, it is true that <span class="math-container">$a_n \leq b... | Kevin Sheng | 150,297 | <p>You are asked to show $a_m \le r \le b_n$. Notice that the indices must be different, the result you give has the same index (i.e. $ a_n
\le r \le b_n$). Given $m,n \in \mathbb{N}$, we have without loss of generality $m\le n$. Then $a_m \le a_n \le b_n$ by your hypothesis, take $r=a_n$ and you are done.</p>
|
27,759 | <p>Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?</p>
<p>Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can... | Jonas Meyer | 1,424 | <p>This would be easy for a diagonal matrix, because $\begin{bmatrix}a & 0\\ 0 & b\end{bmatrix}^3=\begin{bmatrix}a^3 & 0\\ 0 & b^3\end{bmatrix}$, which means that you could just take the cube root of each diagonal entry to solve the problem. While $A$ is not diagonal, it is symmetric and therefore diag... |
276,818 | <p>I am attempting to numerically optimize a function of only 2 parameters <code>k, \[Theta]</code>.</p>
<p>The function is well-defined and the constraints are simple. However, the optimisation keeps returning <code>Indeterminate</code> or <code>not a real number</code> comment over the parameter space, i.e. <code>fun... | Nasser | 70 | <p>You can't obtain the recurrence formula with V 13.1</p>
<p>Looking at "Q&A with Calculus Developers: Live with the R&D team" Nov 30, 2022 at
<a href="https://www.twitch.tv/videos/1666674721" rel="nofollow noreferrer">https://www.twitch.tv/videos/1666674721</a> for V 13.2, I did not see this featur... |
31,539 | <p>I want to learn a bit about Linear Programming. </p>
<p>After some research, I decided to solve the <a href="http://en.wikipedia.org/wiki/Cutting_stock_problem" rel="nofollow">Cutting Stock</a> problem as an example to learn. After doing some more research, I feel like I finally understand Linear Programming enough... | Timothy Chow | 3,106 | <p>I'd recommend you start by consulting the <a href="http://www.neos-guide.org/content/lp-faq" rel="nofollow">Linear Programming FAQ</a>.</p>
<p>As for a solver, I'd recommend <a href="http://www.gnu.org/software/glpk/" rel="nofollow">GLPK</a>, which is free and can handle integer linear programming.</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | alex | 98,428 | <p>Is it correct that your number equals 1 or it equals 2 and {any question she don't know answer for}?</p>
<p>Yep, it's a boring answer, but it always works for such kind of problems.</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | PearsonArtPhoto | 5,101 | <p>Most of the answers seem to rely on relatively complex mathematics, that I couldn't easily do in my head. I'm hoping this is the simpliest answer out there, or at least, one of the simplest.</p>
<p>If I take your number, and subtract 2 from it, then take the reciprocal, is it positive? In other words: $\frac{1}{x-2... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Kostya | 28,158 | <p>How about this one:</p>
<blockquote>
<p>Let's say your number is $n$.<br>
For every even number $x$ ($x>2$) is that true that $x+n$ is representable as a sum of $n$ primes?</p>
</blockquote>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Xiaochuan | 5,476 | <p>The boy gives the three numbers different names: "Yes", "No" and "I don't know". So the question is, "what is the name of your number?"</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | chubakueno | 37,844 | <p>I am thinking of a positive integer. Is your number, raised to my number and then increased in $1$, a prime number?</p>
<p>$$1^n+1=2\rightarrow \text{Yes}$$
$$2^n+1=\text{possible fermat number}\rightarrow \text{I don't know}$$
$$3^n+1=2k\rightarrow \text{No}$$</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Terry Li | 60,889 | <p>Assume the girl's number is X.</p>
<p>The boy asks:</p>
<blockquote>
<p>Is half day before OP posted this question past October X?</p>
</blockquote>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Rigel | 189,549 | <p>Divide the other two numbers left to get a proper fraction. Ask her to subtract 1 from numerator and 2 from denominator. Ask her if the 'number' obtained is greater than 0.</p>
<p>If yes. She should have had 1 and thought of number 1.
If she says she doesn't know. The number she thought of must be 3
If no. It shoul... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Jack D'Aurizio | 44,121 | <p>Let $k$ be your your number. Is there a $3k$-colouring of the elements of $\mathbb{R}^2$ such that points having a unit distance have different colours?</p>
<p><em>Yes</em>$\,\to k=3.\quad$<em>No</em>$\,\to k=1.\quad$<em>I don't know</em>$\,\to k=2.$</p>
<p>Have a look at <a href="https://www.cut-the-knot.org/proo... |
910,414 | <p>My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that "two parallel lines intersect." If you type projective space into Wikipedia, or look it up in a textbook, a lot of t... | Bananeen | 165,356 | <p>The mathematical reason for this is like this:
suppose you look at an infinitely large plane from a point 1 meter above that plane and you have two infinitely long parallel lines drawn on it. Further, suppose the distance between lines is also 1 meter. Then, when you look at these lines right under your point of vie... |
174,339 | <p>This one is somewhat hard to explain, but I'll try my best to.<br>
I'm trying to generate a list of numbers (containing only pi/10, 0, -pi/10). Numbers are randomly selected from these 3, where the probability of getting 0 is always 70%. But the probability of getting pi/10 and -pi/10 depend on the previous outcomes... | mef | 449 | <p>This answer relies on the built-in function $\texttt{DiscreteMarkovProcess}$. In order to use it, the states of the world must be specified. A state is composed of two items: the current value and the most recent nonzero value. (Again I use {-1, 0, 1} for the values instead of {-Pi/10, 0, Pi/10}.) </p>
<p>There are... |
763,381 | <p>There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the ... | JLBarba | 144,734 | <p>I guess the answer is that $f(x)$ has no roots.
<br>Firstly we can see that degree of $f(x)$ must be even.
Why?. Using the left hand side of the definition $dgr(f(x))=2m$.
<br>Then if we had one root $(r_0)$, we could build the increasing sequence $r_{n+1}=r_n^2+1$ of solutions. Which it's a contradiction with the... |
308,014 | <p>First of all, on this Matlab exercise sheet that I am currently working through what does the term 'the lower $2 \times 2$ block' mean in the question below?</p>
<p>$A =
\left[\begin{array}\
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{array}\right]$,</p>
<p>Define a $2 \times 2$ matrix tha... | Bravo | 24,451 | <p>It is clear from the wording; it is the submatrix $\left[\begin{array}\
5 & 6 \\
8 & 9 \\
\end{array}\right]$. In Matlab, you need to say $A(2:3,2:3)$ to get the submatrix.</p>
|
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | paul garrett | 12,291 | <p>First, the <em>formal</em> question "must a function be defined at a point to be continuous there?" has answer "yes", from the definition of "continuity", since the latter definition refers to the value of the function at the point.</p>
<p>But this formal sense of the question does not go far enough, in my opinion.... |
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | Alexander Shamov | 37,460 | <p>Surely, this is not a precise mathematical question - or, at least, as a precise mathematical question it is rather trivial: there are lots of discontinuous functions that are perfectly well-defined everywhere, if you want them to.</p>
<p>However, there is a deeper perspective coming from algebraic geometry and fun... |
867,938 | <p>I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as:</p>
<p>$||x|| = \sqrt{x_1^2 + ... + x_n^2}$</p>
<p>and says:</p>
<p>"The norm is not linear on $R^n$. To inject linearity into the discussion, we introduce the dot ... | Community | -1 | <p>Does</p>
<p>$$\|x+y\|=\|x\|+\|y\|\quad?$$
Find a counterexample and recall the triangle inequality.</p>
|
1,287,692 | <p>To show whether or not the 3 planes
$$x+y-2z=5\tag 1$$
$$x-y+3z=6 \tag2$$
$$x+5y-12z=12 \tag 3$$ all have a common line of intersection.</p>
<p>Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ to get the line $2y-5z=-1$ which is $6y-15z=-3$ , and say that as these aren't the same line, they don't have ... | Dinesh Lama | 349,528 | <p>first by solving 2 planes find y and z, where u have to consider z as t, hence u'l get parametric equation of y and z w.r.t t. now put this value in any plane which will give u parametric equation of x in terms of t only. Now substitute this values in any sphere, than u'll get quadratic equation in terms of t, if it... |
2,768,308 | <p>If there is an easier way to see this I would be thrilled to know that way. I think the trick is to make sure the maximum of $f_n$ occurs to the right of a fixed $x_0$ so that we can avoid the function converging to $\frac{1}{e}$.</p>
<p><strong>Attempt.</strong></p>
<p>For x=1 and x=0, $f_n(x)=0$. Fix an $x_0\in(... | user539887 | 539,887 | <p>Fix $x \in (0,1)$. It suffices to prove that
$$
\lim\limits_{n \to \infty} (n x^n) = 0.
$$
We have
$$
\tag{$*$}
\frac{(n+1) x^{n+1}}{n x^n} = \Bigl( 1 + \frac{1}{n} \Bigr) x,
$$
which is $< 1$ for $n > x/(1-x)$. Consequently, the sequence $(n x^n)$, being eventually decreasing and bounded below (by $0$), co... |
4,584,609 | <p>Which is bigger</p>
<p><span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\sin x}{1+x^2}dx$$</span> or <span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\cos x}{1+x^2}dx~?$$</span></p>
<p>I let <span class="math-container">$x=\frac{\pi}{2}-t$</span> in the second integral, and I obtain this
<span ... | dezdichado | 152,744 | <p>Actually, this is rather crude:
<span class="math-container">$$\int_0^{\frac{\pi}{2}}\dfrac{\sin x}{1+x^2}dx<\int_0^{\frac{\pi}{2}}\dfrac{x}{1+x^2}dx = \dfrac{1}{2}\ln\dfrac{4+\pi^2}{4}\approx 0.62 < 0.72\approx \dfrac{3}{2}\arctan\frac{\pi}{2} - \frac{\pi}{4}=\int_0^{\frac{\pi}{2}}\dfrac{1-x^2/2}{1+x^2}dx <... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | lhf | 589 | <p><a href="http://en.wikipedia.org/wiki/Linear_dynamical_system" rel="nofollow noreferrer">Linear discrete dynamical systems</a>, aka <a href="http://en.wikipedia.org/wiki/Recurrence_relation" rel="nofollow noreferrer">recurrence relations</a>, are best studied in a matrix formulation <span class="math-container">$x_{... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Niel de Beaudrap | 439 | <p>Matrix multiplcation plays an important role in quantum mechanics, and all throughout physics. Examples include the <a href="http://en.wikipedia.org/wiki/Moment_of_inertia#Moment_of_inertia_tensor" rel="noreferrer">moment of inertia tensor</a>, continuous-time descriptions of the evolution of physical systems using ... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Freddie | 11,471 | <p>Hey Alex, a central theme of Machine Learning is about finding structures (preferably linear ones) in the data space; the intrinsic dimentionalities of your observations if you may (see Eigenfaces).</p>
<p>I understand this may not be about matrix multiplication per se; instead, this is about what, many times, happ... |
991,372 | <p>Consider $f_{k}(x)=(\frac{1}{ x}\frac{d}{d x})^k(\frac{x}{\sinh x})$, $x>0$, $k=0,1,2,\cdots.$ Then, Is $f_{k}(x)$ always a bounded function? </p>
<p>The only thing one need to care is the behavior when $x$ is near $0$, and prove it's bounded. I try the case $k=1,2,$ and I think it's true for general k as well.... | Blue | 409 | <p>Consider $A$'s relationship with various circles.</p>
<p>Since $B^\prime$ and $C^\prime$ are points of tangency of lines through $A$ with the incircle, we have
$$|AB^\prime| = |AC^\prime| \qquad (\star)$$
Further, by the <a href="http://en.wikipedia.org/wiki/Power_of_a_point#Theorems" rel="nofollow">"Power of a Poi... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | J. W. Tanner | 615,567 | <p>Here is a different approach.</p>
<p>Modulo <span class="math-container">$7$</span>, there aren't so many cubes, so that can be a good setting to investigate such problems:</p>
<p><span class="math-container">$2^n+1\equiv 2, 3, $</span> or <span class="math-container">$5\pmod7$</span>, but <span class="math-containe... |
1,204,721 | <p>How can I prove that A is non-invertible nxn matrix if sum of all elements in rows of A equal to zero?</p>
<p>$$a_{11} + a_{12} + \cdots + a_{1n} = 0$$</p>
<p>$$a_{21} + a_{22} + \cdots + a_{2n} = 0$$</p>
<p>$$\vdots$$</p>
<p>$$a_{n1} + a_{n2} + \cdots + a_{nn} = 0$$</p>
| DeepSea | 101,504 | <p>The equation $A\cdot X = 0=A\cdot 0$ has a non-zero solution, namely $X = (1,1,..,1)$. This shows that the property that: " $AB = AC \Rightarrow B = C$" fails, thus $A$ is not invertible.</p>
|
1,204,721 | <p>How can I prove that A is non-invertible nxn matrix if sum of all elements in rows of A equal to zero?</p>
<p>$$a_{11} + a_{12} + \cdots + a_{1n} = 0$$</p>
<p>$$a_{21} + a_{22} + \cdots + a_{2n} = 0$$</p>
<p>$$\vdots$$</p>
<p>$$a_{n1} + a_{n2} + \cdots + a_{nn} = 0$$</p>
| Nicolas | 213,738 | <p>Let $A=\left(C_{1},\ldots,C_{n}\right)$
where $C_{j}$
is the $j$ -th column of $A$
. Then your condition is $C_{1}+\ldots+C_{n}=0$
that is $C_{n}=-\left(C_{1}+\ldots+C_{n-1}\right)$
i.e. the columns of $A$
are linearly dependant. Thus the determinant is zero.</p>
|
339,289 | <blockquote>
<p>Prove that if $f$ is defined for $x\ge 0$ by $f(x)=\sqrt x$, then $f$
is continuous at every point of its domain.</p>
</blockquote>
<p>Definition of a continuous function is:</p>
<p>Let $A\subseteq\mathbb{R}$ and let $f:A\to\mathbb{R}$. Denote $c\in A$.</p>
<p>Then $f(x)$ is continuous at $c$ iff... | Ian | 83,396 | <p>user62089's answer is not quite right. At $c \neq 0$, user62089's work proves that $\delta = \varepsilon \sqrt{c}$ is sufficient. At $c=0$ you have a slightly different situation: you want $\delta>0$ so that $\sqrt{x}<\varepsilon$ for $x<\delta$. $\delta=\varepsilon^2$ works for this purpose. </p>
|
1,410,164 | <p>I've found the following identity.</p>
<blockquote>
<p>$$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ </p>
</blockquote>
<p>I could verify it by using CAS, and calculate the integrals in term of <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">exponential</a> and <a ... | Noam Shalev - nospoon | 219,995 | <p>Define $I(a)=\int_0^{\infty} e^{-(x+1)a}\frac{\sin(x)}{x+1}dx$.
Then $\displaystyle I'(a)=-\int_0^{\infty} e^{-(x+1)a}\sin(x)dx=-\frac{e^{-a}}{a^2+1}$, and since $\lim_{a\to \infty} I(a)=0$, we have $\ \displaystyle I(0)=\int_0^{\infty} \frac{\sin(x)}{x+1}=\int_0^{\infty}\frac{e^{-a}}{a^2+1}da=\int_0^1\frac{dx}{1+... |
794,182 | <p>I am trying to find an example of when $(A \bigcup B)^\circ \supset A^\circ \bigcup B^\circ$. Where $^\circ$ denotes the interior of a set. It has been previously shown in <a href="http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf" rel="nofollow">the text</a> that:</p>
<blockquote>
<p>The interior of the set $\... | user139388 | 139,388 | <p>$1 \in R(f)$ if and only if
$$
7x - x^2
= 12 e,
$$
or
$$
x
= \frac{7 \pm \sqrt{49-48 e}}{2}.
$$
Since you are dealing with a function with a real domain, and these values of $x$ do not lie in $\mathbb{R}$, there is no pre-image of $1$.</p>
|
794,182 | <p>I am trying to find an example of when $(A \bigcup B)^\circ \supset A^\circ \bigcup B^\circ$. Where $^\circ$ denotes the interior of a set. It has been previously shown in <a href="http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf" rel="nofollow">the text</a> that:</p>
<blockquote>
<p>The interior of the set $\... | Mikotar | 149,637 | <p>Your problem is that you are confounding the domain of a function with its range. A domain is the set of valid inputs of a function. The range of a function is its set of possible outputs. In your case, that set of inputs requires two things: (i) that $\frac{7x−x^2}{12}$ is positive, and (ii) that $ln(\frac{7x−x^2}{... |
827,899 | <p>I came across this problem on a HackerRank challenge.</p>
<p>The function $f(n)$ is</p>
<ul>
<li>$1$ if $n = 0$</li>
<li>$2f(n - 1)$, if $n$ is odd</li>
<li>$f(n -1) + 1$, if $n$ is even</li>
</ul>
<p>I solved the problem using a recursive function and it worked just well. However, I am assuming that a program wo... | Joel | 85,072 | <p>First I would start by writing out the sequence: $$f(0)=1, f(1) = 2, f(2) = 3, f(3) = 6, f(4) = 7, f(5) = 14, f(6) = 15, ...$$</p>
<p>Now notice that $f(0) = 2 - 1 = 2^{0+1} - 1$, $f(2) = 4-1 = 2^{1+1} - 1$, $f(4) = 8-1 = 2^{2+1} - 1$, and $f(6) = 16 - 1 = 2^{3+1}-1$. So we conjecture that $$f(2n) = 2^{n+1} - 1$$</... |
827,899 | <p>I came across this problem on a HackerRank challenge.</p>
<p>The function $f(n)$ is</p>
<ul>
<li>$1$ if $n = 0$</li>
<li>$2f(n - 1)$, if $n$ is odd</li>
<li>$f(n -1) + 1$, if $n$ is even</li>
</ul>
<p>I solved the problem using a recursive function and it worked just well. However, I am assuming that a program wo... | Jika | 143,855 | <p>I tried to solve it as follow:</p>
<p>For $n$ odd, let $n=2k+1$ for some $k$. Then:
$$
f_n=f_{2k+1}=2f_{2k}=2\left(f_{2k-1}+1\right)=2f_{2k-1}+2=2^2f_{2k-2}+2=\cdots
$$
I continue until $f_0$. And I did the same thing for $n$ even. So I get the following:</p>
<p>Let, $n\div2=p$ and $n\mod2=q=\begin{cases}0\, \text... |
154,818 | <p>Let $G$ be a finitely generated group. The <em>weight</em> $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called the <em>normal rank</em>). Obviously, $d(G^{\operatorname{ab}})\leq w(G) \leq d(G)$, where $d(G)$ is the rank ... | Luc Guyot | 84,349 | <p>The answer is positive for finitely generated soluble groups. (And also for finitely generated simple groups, but it's easy!)</p>
<blockquote>
<p><strong>Claim.</strong> Let <span class="math-container">$G$</span> be a finitely generated soluble group. Then <span class="math-container">$G$</span> has a generating se... |
4,186,303 | <p>Given the following definitions, taken from Wikipedia:</p>
<blockquote>
<p><strong>Support of a function</strong>. When <span class="math-container">$X$</span> is a topological space and <span class="math-container">$f : X \to \mathbb C$</span> is a continuous function, the <em>support</em> of <span class="math-cont... | David C. Ullrich | 248,223 | <p>Say <span class="math-container">$U$</span> is the largest open set in which <span class="math-container">$u$</span> vanishes; let <span class="math-container">$O$</span> be the complement of the support of <span class="math-container">$g$</span>, so <span class="math-container">$O=(g^{-1}(0))^o$</span>.</p>
<p>The ... |
2,465,803 | <p>Given a random permutation <span class="math-container">$\sigma \in S_n$</span> from <span class="math-container">$[n] \to [n]$</span> in a uniform probability space, what is the probability that <span class="math-container">$\sigma $</span> has exactly <span class="math-container">$k$</span> fixed points for a give... | drhab | 75,923 | <p>There are $\binom{n}{k}$ possibilities for the $k$ fixed points.</p>
<p>If they are established then there are $!(n-k)$ <a href="https://en.wikipedia.org/wiki/Derangement" rel="noreferrer">derangements</a> for the other points. </p>
<p>That gives $\binom{n}{k}\left[!(n-k)\right]$ permutations with exactly $k$ fixe... |
2,465,803 | <p>Given a random permutation <span class="math-container">$\sigma \in S_n$</span> from <span class="math-container">$[n] \to [n]$</span> in a uniform probability space, what is the probability that <span class="math-container">$\sigma $</span> has exactly <span class="math-container">$k$</span> fixed points for a give... | Marko Riedel | 44,883 | <p>By way of enrichment here is an alternate formulation using
combinatorial classes. The class of permutations with fixed points
marked is</p>
<p><span class="math-container">$$\def\textsc#1{\dosc#1\csod}
\def\dosc#1#2\csod{{\rm #1{\small #2}}}
\textsc{SET}(\mathcal{U} \times
\textsc{CYC}_{=1}(\mathcal... |
2,307,753 | <p>I know that $-\int \tan(t)dt$ = $\ln |\cos t|$ (letting $C=0$). So I would think that $e^{-\int \tan(t)dt}$ would be equal to $e^{\ln |\cos t|} = |\cos t|$. However, my math textbook and Wolfram Alpha both say that $e^{-\int \tan(t)dt}=e^{\ln (\cos t)} = \cos t$. Why can the absolute value be ignored when taking the... | G.H.lee | 445,037 | <p>(integrating factor : $ \mu (t)$)</p>
<p>Then </p>
<p>$\ln \left | \mu (t) \right | = - \int \tan(t)dt = \ln \left | \cos t \right |$</p>
<p>so $|\mu (t)| =|\cos t|$</p>
<p>It means $\mu (t) =\pm \cos t$</p>
<p>But $ \mu (t)$ should be differentiable. (When $\cos t \neq 0$) </p>
<p>so </p>
<p>Case 1: $\mu ... |
2,307,753 | <p>I know that $-\int \tan(t)dt$ = $\ln |\cos t|$ (letting $C=0$). So I would think that $e^{-\int \tan(t)dt}$ would be equal to $e^{\ln |\cos t|} = |\cos t|$. However, my math textbook and Wolfram Alpha both say that $e^{-\int \tan(t)dt}=e^{\ln (\cos t)} = \cos t$. Why can the absolute value be ignored when taking the... | John Wayland Bales | 246,513 | <p>Your question is "Why can the absolute value be ignored?"</p>
<p>We have</p>
<p>\begin{eqnarray} e^{-\int\tan t\,dt}&=&e^{\ln\vert\cos(t)\vert+c_1}\\
&=&e^{c_1}\vert\cos(t)\vert
\end{eqnarray}</p>
<p>But $e^{c_1}>0$ for all values of $c_1$ and $\vert\cos(t)\vert\ge0$ when in fact solutions such... |
8,764 | <p>Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??</p>
<p>I think the result is
Q[x]/(x) \ otimes_{Q[x]}Q(x) but on the other hand, from another way, I know it should be Q[1/x]/Q.</p>
<p>But ho... | Rado | 1,645 | <p>Isn't $Q[x]/(x)$ just $Q$. In general $Q[x]/(x-\alpha)$ is just evaluation of all polynomials at alpha.</p>
<p>Then localization does nothing as we already have a full field of fractions.</p>
|
1,031,033 | <p>I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ones also seemed easy after I grasped them, so I'm sure that it's just a bit hard for me to get the first thought on the... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> Both sides are multiples of $1+x$.</p>
|
1,031,033 | <p>I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ones also seemed easy after I grasped them, so I'm sure that it's just a bit hard for me to get the first thought on the... | amWhy | 9,003 | <p>$$\begin{align} 1+x^3\geq x+x^2 &\iff x^3-x \geq x^2 - 1\\ \\
& \iff x(x^2-1) \geq x^2 - 1 \quad\forall x\geq 0\end{align}$$</p>
|
1,031,033 | <p>I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ones also seemed easy after I grasped them, so I'm sure that it's just a bit hard for me to get the first thought on the... | Dr. Sonnhard Graubner | 175,066 | <p>we have to prove that<br>
$1+x^3\geq x(1+x)$ for all $x$ with $x>0$,
then we have
$1+x^3-(1+x)x\geq 0$<br>
$(1+x)(1+x^2-x)-x(1+x)\geq 0$
this is equivalent to<br>
$(1+x)(1+x^2-2x)\geq 0$
or
$(x-1)^2(x+1)\geq 0$
which is true.
Sonnhard.</p>
|
246,827 | <p>Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite. </p>
<p>Consider, though, the results of the following numerical experiment. I generated $A$ by letting its eigenvalues be random in $[0,1]$, and selecting its eigenvectors by ge... | Robert Israel | 13,650 | <p><span class="math-container">$\text{tr}(AB+BA) = 2 \operatorname{tr}(A^{1/2} B A^{1/2}) > 0$</span>, so that may produce some bias toward positive eigenvalues. In particular if you generate your "random" matrices in such a way that the eigenvalues of <span class="math-container">$AB+BA$</span> will ten... |
1,415,214 | <p>I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. I've tried solving in several different ways and always got the same answer so I used my graphing calculator to test... | Aaron Maroja | 143,413 | <p><strong>Hint:</strong> Remember that you apply the Fundamental Theorem of Calculus as in </p>
<p>$$\int_{x^2}^{x^8} \sqrt{8t} dt = \int_{c}^{x^8} \sqrt{8t} dt - \int_{c}^{x^2} \sqrt{8t} dt $$</p>
<p>then </p>
<p>$$\frac{d}{dx} \left(\int_{x^2}^{x^8} \sqrt{8t} dt\right) = 8x^7\sqrt{8 x^8} - 2x\sqrt{8 x^2} $$</p>... |
352,243 | <p>In the page 10 of the paper "Filling Riemannian manifolds" by Gromov <a href="https://projecteuclid.org/euclid.jdg/1214509283" rel="noreferrer">(ProjetEuclid link)</a>, the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
<span class="math-container">$$\operatorn... | Max Alekseyev | 7,076 | <p>Yes, such construction is always possible.</p>
<p>Consider two sets of pairs of values:
<span class="math-container">$$\big\{ (2+t,m-t)\quad :\quad t=0\,..\,\lfloor\frac{m-1}{4}\rfloor-1\big\},$$</span>
where differences of elements modulo <span class="math-container">$m$</span> are: <span class="math-container">$2,... |
103,164 | <p>I want to solve an equation which contains an infinite continued fraction $F(n)$. Then I must (obviously) truncate this continued fraction at $n=2000$.</p>
<p>The problem here is that <em>Mathematica</em> does not display the $2000$ terms of this fraction on the screen. The screen closes directly.</p>
<p>Please, h... | J. M.'s persistent exhaustion | 50 | <p>As Mr. Wizard supposed in a comment, one can indeed use <code>ContinuedFractionK[]</code> here:</p>
<pre><code>With[{A = 3., B = 2., x = 0.1},
1/(1 + I A x + ContinuedFractionK[-n^2/(4 n^2 - 1) x^2 A (1 - I 2 B x),
1 + I A x, {n, 2, 2000}])]
0.9197103744410972 - 0.28... |
3,623,724 | <p>The game works as follows: assume that there are two players, 1 and 2. 1 decides to lie or tell the truth. If 1 tells the truth, 2 needs to decide to take her medications or not. Later, in case 2 did not take her medications or 1 lied to her, 2 needs to decide to go to the hospital or not.
If 1 lied to her and she w... | Community | -1 | <p>Yes, you should say what each player should do in every subgame of the game. Hagen is essentially analyzing the strategic form of the game, rather than the extensive form.</p>
<p>In the extensive form, working backwards,</p>
<ul>
<li>If 1 told the truth and 2 did not take the meds, 2 is indifferent between going t... |
11,651 | <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p>
<p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Ca... | Joseph Malkevitch | 1,865 | <p>There are elementary results in mathematics being produced regularly either in the sense that the ideas don't need a lot of background to understand them or sometimes that even the "details" of the proofs are comprehensible. An example, is that of the de Bruijn graph, which has ties to the theory of Eulerian circuit... |
803,488 | <p>Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.</p>
<ul>
<li><p>If you win with Rock, you get \$9. Your opponent loses the \$9.</p>
</li>
<li><p>If you win with Paper, you get \$3. Your opponent loses the \$3.</p>
</li>
<li><p>If you win with Scissors, you get \$5. Your... | Peter Franek | 62,009 | <p>If "optimal" means Nash equilibrium (i.e. a state that is stable wrt. small perturbations of strategies), than it can be computed. If you assume that $x_1$ is the probability of first player to play Rock, $x_2$ his probability to play Scissors and $1-x_1-x_2$ his probability to play Paper, and similarly for $y_i$, t... |
3,162,547 | <p>I am trying to convert instances of nested 'for' loops into a summation expression. The current code fragment I have is:</p>
<pre><code>for i = 1 to n:
for j = 1 to n:
if (i*j >= n):
for k = 1 to n:
sum++
endif
</code></pre>
<p>Basically, the 'if' conditional is c... | Community | -1 | <p>At the basic level, for loops are also rewritable as conditionals. Anyways, on to the meat of the problem. The conditional can be removed if you replace 1 in the previous loop with <code>ceil(n/i)</code>. So it can be rewritten as:</p>
<p><code>for i = 1 to n:
for j = ceil(n/i) to n:
for k = 1 to... |
2,371,668 | <p>For a finite group $G$ and a subgroup $H,$ Lagrange's theorem says that $|G|=|G:H||H|,$ where $|G:H|$ is the number of cosets of $H$ in $G.$<br>
My question is for any subgroup $H$ can we find another subgroup with order $|G:H|$?</p>
| hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>in your inequality, $\ge $ must be $\le $.</p>
<p>For every real $x\ge 0 $ we have
$$ -x^2+2x-1\le 0$$
thus
$$-x+2\sqrt {x}-1\le 0$$</p>
<p>$$\implies 2\sqrt {x}-x\le 1$$
$$\implies \sqrt {x}-\frac {x}{2}\le \frac {1}{2} $$
$$\implies \sqrt {a+b}-\frac {a+b}{2}\le \frac {1}{2} .$$</p>... |
3,488,245 | <p>Define <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> given by <span class="math-container">$f(x)=x^2$</span>. If <span class="math-container">$f^{-1}(x)$</span> is interpreted as a function, it is undefined, since <span class="math-container">$f$</span> is not injective. If <span class="math-cont... | Community | -1 | <p>This is why we include domains and codomains in function definitions. Given an invertible mapping <span class="math-container">$f:A\to B$</span>, the inverse mapping is <span class="math-container">$f^{-1}:B\to A$</span>. Given any mapping <span class="math-container">$f:A\to B$</span>, the preimage mapping is <sp... |
23,378 | <p>Franel uses the convergence of</p>
<p>$ \frac{\zeta(s+1)}{\zeta(s)} = \sum \frac{c(n)}{n^s}$</p>
<p>as an equivalent to the Riemann hypothesis.</p>
<p>Does anybody have a citation for this result and/or hints for computing $c(n)$?</p>
<p>Thanks for any insight.</p>
<p>Cheers, Scott</p>
| François G. Dorais | 2,000 | <p>Since
$$\zeta(s+1) = \sum_{n=1}^\infty \frac{1/n}{n^s}$$
and
$$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}$$
where $\mu$ is the <a href="http://en.wikipedia.org/wiki/M%C3%B6bius_function">Möbius function</a>, we have
$$c(n) = \sum_{d \mid n} \frac{d}{n}\mu(d) = \frac{1}{n}\prod_{p \mid n} (1-p)$$
using... |
3,132,039 | <p>I am trying to create a set of 10-tuples, where each tuple consists of elements taken from a set {defective, nondefective). Each sample must have precisely 1 defective part and 9 nondefective parts. My attempt</p>
<p>Let the sample be a 10-tuple <span class="math-container">$$S:=(X,\Omega,F)$$</span> <span class="m... | Ben Dyer | 164,207 | <p>Left exact on all open sets implies left exact on stalks. This follows from exactness of direct limits for categories of modules, cf. <a href="https://math.stackexchange.com/questions/121122/why-do-direct-limits-preserve-exactness">Why do direct limits preserve exactness?</a>.</p>
|
8,237 | <p>In this example, I want a series of four buttons to change the value of a variable that is used dynamically to drive a plot. I am trying to figure out why using Table around the buttons causes a problem.</p>
<p>This works:</p>
<pre><code>{Button["1", freq = 1], Button["2", freq = 2], Button["3", freq = 3],
Butt... | Mr.Wizard | 121 | <p>As already stated this is an evaluation problem because of the Hold attribute that <code>Button</code> has, and the mechanism of <code>Table</code> which is akin to <code>Block</code>. You need a way to get the value into the expression. In this case I would use <code>Function</code> and <code>Array</code>:</p>
<... |
3,477,152 | <p>I am trying to prove that</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$</span></p>
<p>diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", but I have no clue about how I would quickly prove this.</p>
<p>So f... | Lonidard | 206,444 | <p>The sequence <span class="math-container">$a_n = \frac{n}{\sqrt{n+1}}$</span> is asymptotic to <span class="math-container">$b_n ={\sqrt{n+1}}$</span>:
<span class="math-container">$$
\lim_{n\to \infty} \frac{a_n}{b_n} = \lim_{n\to \infty} \frac{n}{n+1} = 1
$$</span>
What can you say about the following series?
<spa... |
2,434,373 | <p><strong>Question</strong></p>
<blockquote>
<p>Using Calculus, find points <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the parabola <span class="math-container">$y=1-x^2$</span> such that an equilateral triangle is formed by the <span class="math-container">$x$</span>-axis a... | BBSysDyn | 6,786 | <p>Dr. Silver graciously responded and forwarded the paper <em>Monte-Carlo Simulation Balancing</em>, found at</p>
<p><a href="http://www.machinelearning.org/archive/icml2009/papers/500.pdf" rel="nofollow noreferrer">http://www.machinelearning.org/archive/icml2009/papers/500.pdf</a></p>
<p>There is a derivation there... |
2,434,373 | <p><strong>Question</strong></p>
<blockquote>
<p>Using Calculus, find points <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the parabola <span class="math-container">$y=1-x^2$</span> such that an equilateral triangle is formed by the <span class="math-container">$x$</span>-axis a... | A.D | 248,459 | <p>There is a small error in the second to last step:</p>
<p>$\nabla_\theta [ h(a,s,\theta) - log \sum_b h(b,s,\theta) \frac{\exp(h(b,s,\theta)}{\sum_b \exp(h(b,s,\theta))}]$</p>
<p>$\phi(a,s) - \nabla_\theta log \sum_b h(b,s,\theta) \frac{\exp(h(b,s,\theta)}{\sum_b \exp(h(b,s,\theta))}$</p>
<p>$\nabla_\theta \log(\... |
1,661,941 | <p>Evaluate </p>
<p>$\binom{n}0-2\binom{n}1 + 3\binom{n}2 +···+(−1)^n(n+1)\binom{n}n$ </p>
<p>which is the same as:</p>
<p>$\sum_{k=0}^n=(-1)^k(k+1)\binom{n}k$</p>
<p><strong>My attempt:</strong></p>
<p>Using the Binomial Theorem, we get:</p>
<p>$(1+x)^n=\binom{n}0+\binom{n}1x+\binom{n}2x^2+...+\binom{n}nx^n$</p>... | Andreas Caranti | 58,401 | <p>Let $n \ge 1$.</p>
<p>Start with $$(1+x)^n=\binom{n}{0}+\binom{n}{1} x+\binom{n}{2} x^2+ \dots +\binom{n}nx^n.$$</p>
<p>Multiply by $x$ to get
$$
x (1+x)^n=\binom{n}{0} x+\binom{n}{1} x^2+\binom{n}{2} x^3+ \dots +\binom{n}{n} x^{n+1}.
$$
Differentiate to get
$$
(1 + x)^n + n x (1 + x)^{n-1}
=
\binom{n}0+2\binom{n}... |
1,661,941 | <p>Evaluate </p>
<p>$\binom{n}0-2\binom{n}1 + 3\binom{n}2 +···+(−1)^n(n+1)\binom{n}n$ </p>
<p>which is the same as:</p>
<p>$\sum_{k=0}^n=(-1)^k(k+1)\binom{n}k$</p>
<p><strong>My attempt:</strong></p>
<p>Using the Binomial Theorem, we get:</p>
<p>$(1+x)^n=\binom{n}0+\binom{n}1x+\binom{n}2x^2+...+\binom{n}nx^n$</p>... | Brian M. Scott | 12,042 | <p>A combinatorial evaluation is also possible. Let </p>
<p>$$f(n)=\sum_{k=0}^n(-1)^k\binom{n}k(k+1)\;,$$</p>
<p>the sum in question. Then</p>
<p>$$\begin{align*}
f(n)&=\sum_{k=0}^n(-1)^k\binom{n}{n-k}(k+1)\\
&=\sum_{k=0}^n(-1)^{n-k}\binom{n}k(n-k+1)\\
&=(-1)^n\sum_{k=0}^n(-1)^k\binom{n}k(n+1-k)\;,
\end{... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| haqnatural | 247,767 | <p><span class="math-container">$$\int { \frac { 1 }{\cos(x) } } dx=\int { \frac { \cos { x }\, dx }{ \cos ^{ 2 }{ x } } } =\int { \frac { d\left( \sin { x } \right) }{ 1-\sin^{ 2 }{ x } } } =\frac { 1 }{ 2 } \int { \left( \frac { 1 }{ 1-\sin { x } } +\frac { 1 }{ 1+\sin { x } } \right) d\left( \sin { x } \ri... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| user84413 | 84,413 | <p>$\displaystyle\int\frac{1}{\cos x}dx=\int\frac{1}{\sin(x+\frac{\pi}{2})}dx=\int\frac{1}{\sin t}dt\;\;$ with $t=x+\frac{\pi}{2}.\;\;$ Now let $u=\tan\frac{t}{2}$ to get</p>
<p>$\displaystyle\int\frac{1}{\frac{2u}{1+u^2}}\cdot\frac{2}{1+u^2}du=\int\frac{1}{u}du=\ln|u|+C=\ln\big|\tan\left(\frac{x}{2}+\frac{\pi}{4}\rig... |
2,451,281 | <p>$$y=x+\frac{1}{x-4}$$ I tried to find range of this function as below
$$y(x-4)=x(x-4)+1\\yx-4y=x^2-4x+1\\x^2-x(4+y)+(4y+1)=0\\\Delta \geq 0 \\(4+y)^2-4(4y+1)\geq 0 \\(y-4)^2-4\geq 0\\|y-4|\geq 2\\y\geq 6 \cup y\leq2$$ this usuall way. but I am interested in finall answer... is the other idea to find function range ... | Peter Szilas | 408,605 | <p>Let $z:= x-4,$ and $z \ne 0$.</p>
<p>$y = z + 1/z +4$. </p>
<p>1) $z \gt 0:$</p>
<p>AM -GM: </p>
<p>$z+1/z \ge 2.$</p>
<p>Hence: $y \in [6,\infty)$.</p>
<p>2) $ z\lt 0$.</p>
<p>Above reasoning for $-z$ (positive),i.e.</p>
<p>$-z + 1/(-z) \ge 2$,</p>
<p>leads to:</p>
<p>$y \in (-\infty, 2]$.</p>
<p>Altoget... |
728,916 | <p>I have been developing an RPG and I finally want to program a better development curve. I have decided to use the following equations</p>
<p>Linear, Quadratic and Cubic</p>
<p>The Linear one is easy. I have no problems with that. But I'm stuck with the other 2.</p>
<p><strong>Quadratic</strong></p>
<p>The formul... | dtldarek | 26,306 | <p>There are two aspects of your questions:</p>
<ul>
<li>How to do it in an abstract, ideal setting.</li>
<li>How to do it in practice.</li>
</ul>
<p>First, if you have $n$ unknowns, you need $n$ data points (modulo special cases), like pairs $(x,y)$. Then you get a system of <em>linear</em> equations, which you can ... |
233,397 | <p>could anyone please clarify me the meaning of the term 'hypothesis'? </p>
<p>with relation to terms 'reasoning' and 'assumption' ?</p>
<p>Many thanks</p>
| glebovg | 36,367 | <p>A hypothesis, in mathematics is just another word for conjecture, and conjectures are based on heuristic arguments, calculations, similar results etc. They are almost never simply guesses. In spite of induction and computer science tags, an induction hypothesis is different (a little). You try a few cases (typically... |
898,002 | <p>Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ exists iff $f'(c-)=g'(c+)$, where $f'$ and $g'$ are obtained using derivative rules?</p>
<p>This would seem reasonable ... | humble | 169,868 | <p>First you have to find RHD at $x=c$ this can be done by taking $\lim_{x\to a+}f'(x)$ provided</p>
<ol>
<li>function is right continuous at $x=a$</li>
<li>there exists some right neighbourhood where $f(x)$ is differentiable</li>
<li>$\lim_{x\to a+}$ exists it is equal to + or - infinity</li>
</ol>
<p>Similarly find... |
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